The heavy path approach to Galton-Watson trees with an application to Apollonian networks

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by  Luc Devroye

THE HEAVY PATH APPROACH TO GALTON-WATSON TREES WITH AN
APPLICATION TO APOLLONIAN NETWORKS
LUCDEVROYE,CECILIAHOLMGREN,ANDHENNINGSULZBACH
ABSTRACT. We study the heavy path decomposition of conditional Galton-Watson trees. In a
standardGalton-Watsontreeconditionalonitssizen,weorderallchildrenbytheirsubtreesizes,
fromlarge(heavy)tosmall.Anodeismarkedifitisamongthekheaviestnodesamongitssiblings.
Unmarkednodesandtheirsubtreesareremoved,leavingonlyatreeofmarkednodes,whichwecall
7 thek-heavytree. Westudyvariouspropertiesofthesetrees,includingtheirsizeandthemaximal
1 distancefromanyoriginalnodetothek-heavytree. Inparticular,undersomemomentcondition,
0
the2-heavytreeiswithhighprobabilitylargerthancnforsomeconstantc>0,andthemaximal
2
distance from the k-heavy tree is O(n1/(k+1)) in probability. As a consequence, for uniformly

n randomApolloniannetworksofsizen,theexpectedsizeofthelongestsimplepathisΩ(n).
a
J

0
1
1. INTRODUCTION
]
R
We study Galton-Watson trees of size n. More precisely, we have a basic random variable ξ
P
definedby
.
h
P{ξ = i} = p , 0 ≤ i < ∞,
t i
a
m where(p ) isafixeddistribution. Throughoutthepaper,weassumethat
i i≥0
[ ∞
(cid:88)
p = 1, E{ξ} = 1,
1 i
v i=0
7 0 < σ2 := E(cid:8)ξ2(cid:9)−1 < ∞.
2
5
The random variable ξ is used to define a critical Galton-Watson process (see, e.g. [9]). In a
2
0 standard construction, we label the nodes of the Galton-Watson tree in preorder. If ξ ,ξ ,...
1 2
.
1 are independent copies of ξ, then node i has ξi children. Clearly, not all sequences (ξi)1≤i≤n
0 correspondtoatreeofsizen,aswemusthave
7
1
(ξ −1)+...+(ξ −1) ≥ 0 forall 1 ≤ i < n,
: 1 i
v
(ξ −1)+...+(ξ −1) = −1.
i 1 n
X
Foragivensequenceξ ,ξ ,...,wedefinethesizeofatreeT as
r 1 2
a
|T| = min{t ≥ 1 : 1+(ξ −1)+...+(ξ −1) = 0}.
1 t
This is a random Galton-Watson tree. Given |T| = n, T is a conditional Galton-Watson tree.
The family of conditional Galton-Watson trees has gained importance in the literature because it
encompasses the simply-generated trees introduced by Meir and Moon [38], which are basically
orderedrootedtrees(ofagivensize)thatareuniformlychosenfromaclassoftrees. Forexample,
when p = p = 1/4,p = 1/2, the conditional Galton-Watson tree corresponds to a random
0 2 1
Date:January11,2017.
1991MathematicsSubjectClassification. 60J80,60J85,05C80.
Key words and phrases. branching processes, fringe trees, spine decomposition, binary tree, continuum random
tree,Brownianexcursion,Apolloniannetworks.
1
2 L.DEVROYE,C.HOLMGREN,ANDH.SULZBACH
binary tree of size n, also called a Catalan tree. When (p ) is Poisson(1), then we obtain a
i i≥0
randomlabeledrootedtree,alsocalledaCayleytree.
1.1. Encoding ordered rooted trees. We consider two encoding functions for Galton-Watson
treesofsizen. Notethatthesemakeperfectlysenseforanyorderedrootedtreewithnnodes. For
1 ≤ t ≤ n,denotebyd(t)thedepthofthet-thnode,wherenodesarelistedinpreorder. First,we
definetheLukasiewiczpath(S ) byS := 0and
i 0≤i≤n 0
S = (ξ −1)+...+(ξ −1), 1 ≤ i ≤ n.
i 1 i
Of course, we have S = −1 and S ≥ 0 for all 0 ≤ i ≤ n. Second, the depth-first process (or
n i
contour function) (D ) is defined by D = d(f(i)), where (f(i)) denotes the
i 0≤i≤2n−2 i 0≤i≤2n−2
nodevisitedinthei-thstepofthedepthfirsttraversalwithrespecttothepreorder. Bothencodings
areextendedtocontinuousfunctionsontherespectiveintervalsonR+ bylinearinterpolation,see
0
Figure1belowforanexample.
1
S D
t t
2 2
2 3 5
1 1
4 6 7
0 0
0 1 7 t 01 12 t
FIGURE1. Afiniterootedtreeofsize7withlabelsgivenbythepreorder. Second
andthirdpictureshowthecorrespondingLukasiewiczpathanddepth-first-search
process.
1.2. ThelocalandtheglobalpictureofGalton-Watsontrees. ThehistoryofconditionalGalton-
Watsontreesisquiterich. Tworesultsstandoutthatencapsulateourunderstanding:
(cid:16) (cid:17)
σD
(i) The process t(√2n−2) tends in distribution to a standard Brownian excursion,
2 n
0≤t≤1
andtheconditionaltreetendsinsomesensetotheso-calledcontinuumrandomtree. This
celebrated result goes back to a series of papers by Aldous [2, 3, 5]. See also Le Gall
[35]andMarckertandMokkadem[37]foradiscussionofconvergenceofbothencoding
processes. This implies that the height H of the conditional Galton-Watson tree, where
n
H = max d(t),satisfies
n 1≤t≤n
σH
n d
√ −→ H ,
∞
2n
d
whereH hasthethetadistributionand−→denotesconvergenceindistribution. Thatis,
∞
(cid:26) (cid:27) ∞
(1) lim P σ√Hn ≤ x = (cid:88) (1−2j2x2)exp(−j2x2), x > 0.
n→∞ 2n
j=−∞
Inthisgenerality,thislimittheoremgoesbacktoKolchin[34,Theorem2.4.3]. Inthecase
of Cayley trees, (1) had already been discovered by Re´nyi and Szerekes [40] and for full
binarytrees,thatisp = p = 1/2,byFlajoletandOdlyzko[26].
0 2
THEHEAVYPATHAPPROACHTOGALTON-WATSONTREES 3
Moreover, there are universal upper bounds that will be useful for this paper: there
existsδ ∈ (0,σ2/2],suchthat
(cid:26) (cid:27)
H
(2) supP √n ≥ x ≤ exp(−δx2), x > 0.
n
n≥1
ThisisTheorem1.2inAddario-Berry,DevroyeandJanson[1].
(ii) As n grows large, T can be thought of as a long spine with offspring defined as follows:
First construct an infinite sequence ζ ,ζ ,... drawn from the distribution (ip ) , also
1 2 i i≥0
called the size-biased distribution. Associate ζ with the i-th node on an infinite path. To
i
every node i on the path assign (ζ − 1) children off the path, and make each child the
i
root of an independent (unconditional) Galton-Watson tree. Finally permute all children
ofeverynodeontheinfinitespine. Thisinfiniteso-calledsize-biasedGalton-Watsontree
isthescalinglimitofconditionalGalton-Watsontreesasn → ∞inamuchdifferentsense
than in (i). The decomposition is called the spine decomposition. The construction goes
back to Kesten [33]. Compare also Lyons, Pemantle and Peres [36], Aldous and Pitman
[6,Section2.5]andJanson[29,Section7].
Let us stress that the two pictures drawn focus on very different aspects of the trees. Aldous’
theory leading to the continuum random tree describes the global structure and is useful in the
analysis of the tree height, diameter or the depth of a uniformly chosen node. The convergence
result constitutes an invariance principle: in analogy to the central limit theorem for independent
and identically distributed summands, it only relies on the second moment of the offspring dis-
tribution. This also means that more local information, that is, quantities which scale on order
√
smallerthan n,cannotbestudiedbythismethod.
The picture drawn in (ii) is local: the conditional Galton-Watson tree converges locally, in the
senseofAldous-Steele[7](sometimesalsoreferredtoasBenjamini-Schrammconvergence[10]),
totheinfinitesize-biasedtree. Tobemoreprecise,itstatesthat,foranyfixedk ≥ 1,theconditional
Galton-Watsontreerestrictedtonodesofdistanceatmostkfromtherootconvergesasn → ∞in
distributiontotherestrictedobjectsampledfromtheinfinitesize-biasedtree.
ThepresentpaperlooksatalessnaturaldecompositionoftheconditionalGalton-Watsontree,
butonethathasfar-reachingapplicationsincomputerscienceandthestudyofrandomnetworks,
moreprecisely,randomApolloniannetworks.
1.3. Heavysubtreesandmainresults. Onecanreorderallsetsofsiblingsbysubtreesize,from
large to small, where ties are broken by considering the preorder index. For a node v in the
(conditionalornot)Galton-WatsontreeT,wedenotebyρ therankinitsordering(forexample,
v
ρ = 1meansthatv hasthelargestsubtreeamongitssiblings). LetA = (v ,...,v = v)be
v v 1 d−1
thesequenceofancestorsofv,startingattherootandendingatvifvisatdistanced−1fromthe
root. Wedefinethemaximalrank
ρ∗ = max(ρ ,...,ρ ).
v v1 vd−1
No rank is defined for the root. For fixed integer k, we define the k-heavy Galton-Watson tree as
the tree formed by the root and {v ∈ T : ρ∗ ≤ k}, where T is the conditional Galton-Watson
v
tree. Thek-heavy tree has nodes of degreek or less. Fork = 1, we obtain a path, which we call
theheavypath—justfollowthepathfromtherootdown,alwaysgoingtothelargestsubtree. Itis
interestingthatthelengthL oftheheavypathhasadifferentasymptoticdistributionalbehaviour
n
than H . Clearly, L ≤ H , but L is neither too small nor too close to H . In Section 6 we
n n n n √ n
discussdistributionalconvergenceofL / nandstudythetailbehaviouroftherandomvariable
√ n
L / n near 0 in more detail. We note that it grows more slowly than any polynomial but much
n
4 L.DEVROYE,C.HOLMGREN,ANDH.SULZBACH
fasterthanthethetalaw(see(1)). Asopposedtothek-heavytreesfork ≥ 2, theheavypathcan
bestudiedusingtheglobalpicture(i)sketchedabove,anditsscalinglimithasarepresentationin
termsofaBrownianexcursion(orthecontinuumrandomtree).
Our main interest, though, is the study of the case k = 2, the 2-heavy Galton Watson tree. In
Section 4, we show that it captures a huge chunk of the Galton-Watson tree: by Theorem 4, if
E(cid:8)ξ5(cid:9) < ∞,thenthereexistsaconstantc > 0suchthat
(3) lim P{Sizeofthe2-heavytree ≥ cn} = 1.
n→∞
SincethenumberofnodesofdegreeiinaconditionalGalton-Watsontreeisinprobabilityasymp-
totictonp ,itiseasytoseethatthesizeofthe2-heavytreecannotbemorethan
i
 
(cid:88)
n1− (i−2)pi+o(1),
i≥3
so that there is no hope of replacing cn by n − o(n) in (3). In fact, we believe that the size of
the2-heavytreesatisfiesalawoflargenumberswhenrescaledbyn−1 asn → ∞withalimiting
constantdependingonthedistributionofξ.
Finally, we also study the maximal distance to the k-heavy trees. For a proper set of nodes,
A ⊆ {1,...,n},wecallthemaximaldistancetoA
maxmindist(v,w),
v∈/A w∈A
where dist(·,·) refers to path distance. The maximal distance to the k-heavy tree measures to
some extent how pervasive the k-heavy tree is. In Section 5, we show that, under appropriate
momentconditionsonξ,thedistanceisinprobabilityΘ(n1/(k+1)). Infact,wealsoshowthatthis
isoptimalinthesensethat,everyk-arysubtreeleavesoutnodesofdistanceordern1/(k+1) away.
1.4. Apollonian networks. In 1930, Birkhoff [16] introduced a model of a planar graph that
becameknownasanApolloniannetwork,anamecoinedbyAndradeetal.[8]in2005. Suggested
as toy models of social and physical networks with remarkable properties, they are recursively
defined by starting with three vertices that form a triangle in the plane. Given a collection of
trianglesinatriangulation,chooseone(eitheratrandom,orfollowinganalgorithm),placeanew
vertexinitscenter,andconnectitwiththethreeverticesofthetriangle. So,ineachstep,wecreate
threenewedges,onenewpoint,andthreenewtriangles(whichreplaceanoldone). Afternsteps,
we have 3+n vertices, and 3+3n edges in the graph. This is an Apollonian network. One can
alsodefineadualtree: startwiththeoriginaltriangleastherootofatree. Inatypicalstep,select
a leaf node of the tree (which corresponds to a triangle) and attach to it three children. This tree
hasaone-to-onerelationshipwiththeApolloniannetwork. Ithas1+2nleaves(afternsteps)and
1+3nvertices. SeeFigure2foranillustration.
A frequently studied (see Zhou, Yan and Wang [43]) random Apollonian network is one in
whicheachtriangle(inthenetwork)—or,equivalently,eachleafinthetree—ischosenuniformly
atrandomforsplitting,leadingtoaso-calledsplittree[20]. (Moreprecisely,weobtainarandom
ternary increasing tree, a variant of the much studied random binary search tree.) Its height is
bounded almost surely by clogn for a suitable constant c > 0 [17]. More importantly, one is
interested in the longest simple path in the Apollonian network. (A simple path in a graph is a
path which visits every vertex at most once.) Calling its length L , its asymptotic behaviour is
n
still not well understood today. Takeo [41] erroneously claimed that Apollonian networks have a
Hamiltonian cycle (and thus, L = n−1), but the so-called Goldner-Harary graphs invented by
n
Gru¨nbaumin1967[28]formjustoneofmanypossiblecounterexamples. FriezeandTsourakakis
THEHEAVYPATHAPPROACHTOGALTON-WATSONTREES 5
FIGURE 2. Apollonian network of size 3 with evolutionary tree. Leaves are
drawninred.
[27] conjectured that for the random Apollonian network of Zhou, Yan and Wang, L ≥ cn for
n
someconstantc > 0withprobabilitytendingtoone. ThiswasdisprovedbyEbrahimzadehetal.
[24] who showed that, with high probability, L = o(n). They also provided a lower bound of
n
Ω(n0.88) for E{L }. Very recently, Collevecchio, Mehrabian and Wormald [19] proved that L
n n
iswithhighprobabilityatmostn1−ε whereεcanbechosen4×10−8.
Iftherandommodelischanged,andwegeneratearandomorderedtreeofsize1+3ninwhich
each non-leaf node has three children, such that all trees are equally likely, then this corresponds
toaconditionalGalton-Watson tree(ofsize1+3n)withp = 2/3,p = p = 0andp = 1/3.
0 1 2 3
Furthermore, it is easy to verify that the length of the longest simple path L is bounded from
n
below by the size of any binary subtree embedded in the Galton-Watson tree. In particular, it is
largerthanthesizeofthe2-heavytree. Therefore,thereexistsc > 0suchthat
(4) lim P{L ≥ cn} = 1.
n
n→∞
Thus,forthisrandommodel,FriezeandTsourakakis’conjectureiseasilysettledbystudyingthe
2-heavytree. Thiswastheinitialmotivationofthepresentpaper.
√
1.5. Notation. Throughout the paper, we use h = gcd{i : p > 0,i > 0}, α = h/(σ 2π),
i
I = {n ≥ 1 : P{S = −1} > 0}, and, for n ∈ I, I = {1 ≤ k ≤ n : P{S = 0} > 0}.
n n n−k
FromBe´zout’slemma,itfollowsthatI = (Nh+1)\AforsomefinitesetA ⊆ N. Intheremainder
ofthepaper, wewriteT forarealizationoftheunconditionalGalton-Watsontreeandτ ,n ∈ I,
n
for T conditional on having size n. (T and τ are considered as graphs, where τ has vertex set
n n
[n] := {1,...,n}.) We introduce the following terminology: for v ∈ [n], let ξ(v) be the number
of children of v, N(v) be the size of the subtree rooted at v, H(v) be the height of the subtree
rooted at v, N (v) be the size of the i-th largest subtree rooted at v, abbreviating N (v) = 0 if
i i
i > ξ(v),andN (v) = N (v)+N (v)+...Weusethenotationξ ,N,H,N andN when
i+ i i+1 (cid:15) i i+
referringtotherootnode. Finally,fork ≥ 1,letZ = |{v ∈ [n] : N(v) = k}|. Westressthat,in
k
ordertoincreasereadability,weoftenomittoindicatetheparameterninthenotation.
In Sections 2 – 6, Appendices A and B, all constants except c,c ,c ,...,C,C ,C ,... carry
1 2 1 2
fixedvalues. Thevaluesofconstantsusedmultipletimesmayvarybetweentworesultsorproofs
but not within. Here, constants C,C ,C ,... > 0 are meant to carry large values, whereas
1 2
c,c ,c ,... > 0 are typically small. Appendix D can be read independently of the remainder
1 2
ofthework.
6 L.DEVROYE,C.HOLMGREN,ANDH.SULZBACH
2. PRELIMINARY RESULTS AND FRINGE TREES
Let us start by recovering some classical results which have proved fruitful in the analysis of
conditional Galton-Watson trees. Recall the following well-known identity going back to Dwass
[23](comparealsoJanson[29,Theorem15.5]andthediscussiontherein),
P{S = −1}
n
(5) P{|T| = n} = .
n
Moregenerally,forindependentcopiesT ,T ,...ofT,
1 2
k
(6) P{|T |+...+|T | = n} = P{S = −k}.
1 k n
n
Inthiscontext,weciteaclassicalresultforsumsofindependentintegerrandomvariablesapplied
tothesequenceS . ByPetrov[39,TheoremVII.1]orKolchin[34,Theorem1.4.2],asn → ∞,
n
(7) x∈sZuhp−n(cid:12)(cid:12)(cid:12)(cid:12)P{Sn = x}− √αn exp(cid:18)−2σx22n(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = o(n−1/2).
√
Inparticular,forx = o( n)withx ∈ Zh−n,asn → ∞,
α
P{S = x} ∼ √ .
n
n
Similarly,asn → ∞,n ∈ Nh+1,
α
(8) P{S = −1} ∼ √ .
n
n
Bysummation,using(5)and(8),ast → ∞,
2α
(9) P{|T| ≥ t} ∼ √ .
h t
ThestudyofthesequenceZ ,k ≥ 1iscloselyrelatedtotheanalysisofarandomfringesubtree
k
τ∗, a subtree of τ rooted at a uniformly chosen node. The study of fringe subtrees was initiated
n n
byAldous[4],whoshowedthat,underourconditions,
(10) τ∗ −d→ T.
n
In particular, E{Z }/n → P{|T| = k} as n → ∞,n ∈ Nh+1 for k ∈ I fixed. Bennies and
k
Kersting[11,Theorem1]generalized(10)tooffspringdistributionswithinfinitevariance. Janson
[29, Theorem 7.12] obtained a quenched version of the result stating that, conditional on the tree
τ , the (random) distribution of τ∗ converges in probability to the (deterministic) distribution of
n n
T. More recently, Janson [30, Theorem 1.5] obtained finer results on subtree counts in τ , in
n
particular estimates and asymptotic expansions for the variance and a central limit theorem. We
summarize special cases of his results in the following proposition. The exact expressions for
meanandvariancearecontainedin[30,Lemma5.1]and[30,Lemma6.1]. Theuniformestimate
onthevariance(12)followsfrom[30,Theorem6.7].
Proposition1(Janson[30]). Letn ∈ I and1 ≤ k ≤ n. Then,
nP{S = −1}P{S = 0}
k n−k
(11) E{Z } = ,
k
kP{S = −1}
n
and,for1 ≤ k ≤ (n−1)/2,
n(n−2k+1)P{S = −1}2P{S = 1}
k n−2k
E{Z (Z −1)} = ,
k k k2P{S = −1}
n
THEHEAVYPATHAPPROACHTOGALTON-WATSONTREES 7
whileE{Z (Z −1)} = 0fork > (n−1)/2. Forfixedk ∈ I,asn → ∞,n ∈ Nh+1,
k k
Var(Zk) → θ2, θ2 = P{|T| = k}(cid:2)1+P{|T| = k}(1−2k−σ−2)(cid:3) > 0.
n
and,
Z −nP{|T| = k}
k √ −d→ N(0,θ2),
n
whereN(0,θ2)denotesanormalrandomvariablewithvarianceθ2 andmean0.
Finally,uniformlyin1 ≤ k ≤ n,asn → ∞,n ∈ Nh+1,
(12) Var(Z ) = O(n).
k
Itfollowsfrom(8)that,asn → ∞andk = o(n)withn ∈ Nh+1,k ∈ I ∩I,inprobability,
n
Z
k
→ 1.
E{Z }
k
Furthermore,ifadditionallyk → ∞,then
αn
(13) E{Z } ∼ .
k k3/2
Manyargumentsinthismanuscriptrelyonboundsonthemeansuchasthosegivenbelow.
Corollary1. Thereexistsaconstantn ≥ 1suchthat,foralln ≥ n ,n ∈ I,k ∈ I ,
0 0 n
 √
4 2α(n−k)−1/2 forn/2 ≤ k ≤ n−n0,
 √
E{Z } ≤ 2 2αnk−3/2 forn ≤ k ≤ n/2,
k 0

(cid:98)n/k(cid:99) for1 ≤ k ≤ n.
Similarly,thereexistconstantsn ≥ 1andς > 0,suchthat,foralln ≥ n ,n ∈ I,k ∈ I ∩I,
1 1 n
(cid:40)
αnk−3/2/2 forn ≤ k ≤ n/2,
1
E{Z } ≥
k
ςn for1 ≤ k ≤ n .
1
Proof. Byanapplicationof(8)to(11),thereexistsn ≥ 1,suchthat,foralln ≤ k ≤ n−n ,
0 0 0
√
(cid:40)
E{Zk} ≤ 2αk−3/2(cid:114) n ≤ 2 √2αk−3/2 forn0 ≤ k ≤ n/2,
n n−k 4√2α forn/2 ≤ i ≤ n−n0.
n n−k
This shows the first two upper bounds. The third follows immediately from the deterministic
boundZ ≤ (cid:98)n/k(cid:99). Thefirstlowerboundfollowsanalogously. Thesecondlowerboundfollows
k
from(5)and(13),since,fork ∈ I,wehaveE{Z }/n → P{|T| = k} = P{S = −1}/k. (cid:3)
k k
Corollary2. ThereexistsauniversalconstantC > 0suchthat,forallM ≥ n andn ≥ M,n ∈
0
I,withn asinCorollary1,wehave
0
(cid:88)n logM √
E{Z }logk ≤ Cn √ +(2+4 2α)n3/4logn.
k
M
k=M
8 L.DEVROYE,C.HOLMGREN,ANDH.SULZBACH
Proof. Byapplicationsoftheupperboundintheprevioustheorem,wehave
(cid:98)(cid:88)n/2(cid:99) √ (cid:98)(cid:88)n/2(cid:99) logM
E{Z }logk ≤ 2 2αn k−3/2logk ≤ Cn √ ,
k
M
k=M k=M
√ √
(cid:98)n(cid:88)− n(cid:99) logn (cid:98)n(cid:88)− n(cid:99) √ √
E{Zk}logk ≤ (cid:112) √ 4 2α ≤ 4 2αn3/4logn,
n−n+ n
k=(cid:100)n/2(cid:101) k=(cid:100)n/2(cid:101)
(cid:88)n (cid:88)n 1 √
E{Z }logk ≤ nlogn ≤ 2 nlogn.
k
k
√ √
k=(cid:100)n− n(cid:101) k=(cid:100)n− n(cid:101)
Theclaimfollowsbysummingthethreeterms. (cid:3)
3. SUBTREES OF THE ROOT: LOCAL CONVERGENCE
We want to understand the properties of the subtree sizes of a node in a Galton-Watson tree
conditional on having size n when these trees are ordered from large to small. This section has
keyinequalitiesthatwillbeneededthroughoutthepaper.
A formulation of the local convergence result discussed in Section 1.2 (ii) is given in the next
propositionwhichisequivalenttoLemma1inDevroye[21]. (Theconvergenceofξ hadalready
(cid:15)
been obtained by Kennedy [32].) We include the short proof for the sake of completeness. Here,
byS ,wedenotethesetofnon-negativeintegervaluedsequencesx ,x ,...withx ≥ x ≥ ...
↓ 1 2 1 2
and only finitely many non-zero elements. Note that S is countable. For k ≥ 1 and 1 ≤ i ≤ k,
↓
and real-valued random variables X ,...,X , denote by X the (k−i+1)-st order statistic.
1 k (i:k)
(ForrandomtreesT ,...,T ,wesimplifythenotationandwrite|T |forthesizeofi-thlargest
1 k (i:k)
tree.)
Proposition2. Letζ havethesize-biaseddistribution(ip ) . Then,asn → ∞,n ∈ Nh+1,in
i i≥0
distributiononS ,
↓
(N ,N ,...) → (|T |,...,|T |,0,0,...),
2 3 (1:ζ−1) (ζ−1:ζ−1)
whereT ,T ,...,ζ areindependent. Indistributionandinmean,ξ → ζ,wherewerecallthatξ
1 2 (cid:15) (cid:15)
is the number of children of the root of τ . The convergence is with respect to the k-th moment if
n
andonlyifE(cid:8)ξk+1(cid:9) < ∞.
Proof. Let k ,k ,... ∈ S with k ∈ Nh,1 ≤ i ≤ (cid:96) − 1, k > 0,k = k = ... = 0
1 2 ↓ i (cid:96) (cid:96)+1 (cid:96)+2
and p > 0. Let y ,...,y be the different values among k ,...,k and α ,...,α be their
(cid:96)+1 1 m 1 (cid:96) 1 m
multiplicities. WithC = (cid:0) (cid:96) (cid:1),
α1,...,αm
(cid:96)
(cid:8) (cid:9) (cid:89)
P (|T |,...,|T |,0,0,...) = (k ,k ,...) = Cp ((cid:96)+1) P{|T| = k }.
(1:ζ−1) (ζ−1:ζ−1) 1 2 (cid:96)+1 i
i=1
Similarly,foralln ∈ I withn > 1+k +(cid:80)(cid:96) k ,
n 1 i=1 i
(cid:110) (cid:111)
(cid:96) P |T| = n−1−(cid:80)(cid:96) k
(cid:89) j=1 j
P{(N ,N ,...) = (k ,k ,...)} = Cp ((cid:96)+1) P{|T| = k } .
2 3 1 2 (cid:96)+1 i
P{|T| = n}
i=1
The distributional convergence in S follows since the fraction in the last display turns to one as
↓
n → ∞. SinceS iscountable,thefunctionf : S → N,f(x ,x ,...) = min{k ≥ 1 : x = 0}
↓ ↓ 1 2 k
THEHEAVYPATHAPPROACHTOGALTON-WATSONTREES 9
iscontinuous,andwededuceξ → ζ indistribution. Furthermore,fork ∈ Nh,using(5)and(6),
(cid:15)
P{|T |+...+|T | = n−1}
1 k
P{ξ = k} = p
(cid:15) k
P{|T| = n}
(cid:18) (cid:19)
1 P{S = −k}
n−1
= kp 1− .
k
n P{S = −1}
n
Sincethefractionisuniformlyboundedink,ξk isuniformlyintegrableifζk isintegrable. Finally,
(cid:15)
if(cid:80) (cid:96)k+1p = ∞,thenE(cid:8)ξk(cid:9) → ∞byFatou’slemma. Thisconcludestheproof. (cid:3)
(cid:96)≥1 (cid:96) (cid:15)
We are interested in tail bounds on N ,k ≥ 2. The order is suggested by the behavior of the
k
limitingrandomvariable.
Proposition3. Letk ≥ 1andassumethatT ,T ,...,ζ areindependent.
1 2
(i) IfE(cid:8)ξk+1(cid:9) < ∞,then,ast → ∞,
(14) P(cid:8)|T | ≥ t(cid:9) = O(t−k/2).
(k:ζ−1)
(cid:80)
(ii) If p > 0,then,ast → ∞,
(cid:96)≥k+1 (cid:96)
P(cid:8)|T | ≥ t(cid:9) = Ω(t−k/2).
(k:ζ−1)
(iii) Finally,ifE(cid:8)ξk+1(cid:9) = ∞,then
lim tk/2P(cid:8)|T | ≥ t(cid:9) = ∞.
(k:ζ−1)
t→∞
Proof. Wehave
(cid:18) (cid:19)
(cid:8) (cid:9) (cid:88) (cid:96)
P |T | ≥ t ≤ p ((cid:96)+1) P{|T | ≥ t,...,|T | ≥ t}.
(k:ζ−1) (cid:96)+1 k 1 k
(cid:96)≥k
By(9),theright-handsideisasymptoticallyequivalentto
(cid:18) 2α (cid:19)k(cid:88) (cid:18)(cid:96)(cid:19)
√ p ((cid:96)+1) .
(cid:96)+1
h t k
(cid:96)≥k
SinceE(cid:8)ξk+1(cid:9) < ∞,thetermisofordert−k/2. For(ii),choose(cid:96) ≥ k withp > 0. Then,
(cid:96)+1
(cid:8) (cid:9) (cid:18) 2α (cid:19)k
P |T | ≥ t ≥ p ((cid:96)+1)P{|T | ≥ t,...,|T | ≥ t} ∼ √ p ((cid:96)+1).
(k:ζ−1) (cid:96)+1 1 k (cid:96)+1
h t
Again,therighthandsideisofordert−k/2.
For(iii),sinceE(cid:8)ξk+1(cid:9) = ∞,foranyC > 0,findK sufficientlylargesuchthat(cid:80)K p ((cid:96)+
(cid:96)=k (cid:96)
1)(cid:0)(cid:96)(cid:1) ≥ C. Then
k
(cid:8) (cid:9) (cid:8) (cid:9)
P |T | ≥ t ≥ P |T | ≥ t,|T | < t
(k:ζ−1) (k:ζ−1) (k+1:ζ−1)
(cid:18) (cid:19)
(cid:88) (cid:96)
= p ((cid:96)+1) P{|T| ≥ t}kP{|T| < t}(cid:96)−k
(cid:96)+1
k
(cid:96)≥k
≥ CP{|T| ≥ t}kP{|T| < t}K.
As t → ∞, using (9), the right hand side is equivalent to C(2αh−1)kt−k/2. As C was chosen
arbitrarily,thefinalassertionofthepropositionfollows. (cid:3)
ThenexttworesultsareprovedinAppendixA.
10 L.DEVROYE,C.HOLMGREN,ANDH.SULZBACH
Theorem1. Letk ≥ 2andE(cid:8)ξk+1(cid:9) < ∞. Then, thereexistsaconstantβ > 0, suchthat, for
k
allt ≥ 1,n ∈ I,
(15) P{N ≥ t} ≤ β t(1−k)/2.
k k
If E(cid:8)ξ(3k+1)/2(cid:9) < ∞, a corresponding bound holds for P{N ≥ t} with β replaced by
k+ k
(cid:8) (cid:9)
some larger constant β . Similarly, bounds of the same form are valid for E ξ 1 if
k+ (cid:15) {N ≥t}
k
E(cid:8)ξk+2(cid:9) < ∞,andforE(cid:8)ξ 1 (cid:9)ifE(cid:8)ξ(3k+3)/2(cid:9) < ∞.
(cid:15) {N ≥t}
k+
Remark. TheproofofTheorem1shows thefollowingstrongerresult: fork ≥ 2, thereexists
aconstantC > 0suchthat,foralln ∈ I,(cid:96) ≥ k andt ≥ 1,
(16) P{N ≥ t,ξ = (cid:96)} ≤ Cp (cid:96)k+1t(1−k)/2.
k (cid:15) (cid:96)
Lemma1belowistheonlyresultinthisworkthatrequiresthisstrongerbound.
d
Remark. SinceN −→ T andthemomentconditiononthisrandomvariableinorder
k (k−1:ζ−1)
tohavetailsdecayingasin(14)istight,itisreasonabletoconjecturethatatailboundsuchas(15)
holdsifandonlyifE(cid:8)ξk(cid:9) < ∞. TheboundspresentedinAppendixAaresufficienttoshowthat
thelatterisindeednecessary: ifE(cid:8)ξk(cid:9) = ∞,thenaboundoftheform(15)isnotvalid. (Aproof
ofthisclaimisgiveninAppendixA.)
Theorem 2. Let k ≥ 2 and (cid:80) p > 0. Then, there exist constants β∗ > 0,s > 0 and
(cid:96)≥k (cid:96) k k
n = n (k) ≥ 1,suchthat,foralln ≥ n ,n ∈ I,and1 ≤ t ≤ n/k−s ,
2 2 2 k
P{N ≥ t} ≥ β∗t(1−k)/2.
k k
From Theorems 1 and 2 we deduce the following corollary using the well-known formula
(cid:82)∞
E{X} = P{X > t}dtforanon-negativerandomvariableX.
0
Corollary3. Asn → ∞,n ∈ Nh+1,
√ √
(i) ifE(cid:8)ξ3(cid:9) < ∞,thenE{N } = Θ( n)andE(cid:8) N (cid:9) = Θ(logn),
2 √ 2
(ii) ifE(cid:8)ξ7/2(cid:9) < ∞,thenE{N } = Θ( n),
2+
(iii) ifE(cid:8)ξ4(cid:9) < ∞,thenE{N } = O(logn),
3
(iv) ifE(cid:8)ξ5(cid:9) < ∞,thenE{N } = O(logn)andE{N } = O(1),and
3+ 4
(v) ifE(cid:8)ξ13/2(cid:9) < ∞,thenE{N } = O(1).
4+
(cid:80)
If p > 0,thenbig-O in(iii)canbereplacedbyΘ.
(cid:96)≥3 (cid:96)
4. THE2-HEAVY TREE
Let T be a finite ordered rooted tree with vertex set V(T). Its root is labeled (cid:15). As in Section
1.3, to each node v ∈ V(T), v (cid:54)= (cid:15), we assign the rank ρ where ρ = i if its subtree is the
v v
i-th largest among all the subtrees rooted at its siblings. Ties are broken by the original order in
the tree. If v has distance k ≥ 1 from (cid:15), let v := (cid:15),v ,..., v ,v = v be the nodes on the
0 1 k−1 k
path connecting the root to v where v has depth i. The path from (cid:15) to v has nodes of indices
i
ρ ,...,ρ = ρ . Itiscalledtheindexsequenceofv anddenotedbyκ(v). Wedefineκ((cid:15)) = ∅
v1 vk v
astheemptyword. Itisconvenienttoborrowsomenotationfromtheoreticalcomputersciencefor
sequencesofintegers: {i ,...,i }denotesonesymbolfromtheset{i ,...,i }andA∗denotesa
1 k 1 k
sequenceofarbitrarylength(even0)drawnfromA ⊆ N. WedefinethesetofnodesV satisfying
a sequence as the collection of all nodes in the tree have index sequences belonging to a set of
sequences. For example, V(1∗) := V({1}∗) is the set of nodes in T that have all their ancestors
anditselfofindex1andtheroot. Ofcourse,thenodesinV(1∗)formtheheavypath. Furthermore,
we recover the k-heavy tree V({1,...,k}∗) of T by removing from T all nodes of index larger