Extremal values in recursive trees via a new tree growth process

0.56 MB·

by  Laura Eslava

EXTREMAL VALUES IN RECURSIVE TREES VIA A NEW TREE
GROWTH PROCESS
LAURAESLAVA
7
Abstract. We give convergence rates on the number of vertices with degree at least
1
clnn, c ∈ (1,2), in random recursive trees on n vertices. This allows us to extend
0
the range for which the distribution of the number of vertices of a given degree is well
2
understood.

n Conceptually, the key innovation of our work lies in a new tree growth process
a ((Tn,σn),n≥1)whereTnisarootedlabeledtreeonnverticesandσnisapermutation
J of the vertex labels. The shape of Tn has the same law as that of a random recursive

6 tree. Interestingonitsownright,thisprocessobtainsTn fromTn−1 byaprocedurewe
call Robin-Hood pruning, which attaches a vertex labeled n to Tn−1 and rewires some

] of the edges in Tn−1 towards the newly added vertex. Additionally, ((Tn,σn),n ≥ 1)
R canbeunderstoodasanewcouplingofallfiniteKingman’scoalescents.
P
.
h
t
a 1. Introduction
m
[ In a paper of 1970 [16], Na and Rapoport presented the problem of modeling how the

structure of networks (as sociograms, communication and acquaintance networks) emerge
1
throughtime. Theyconsideredtwocases. First,aclassof‘growing’ treeswhoseconstruction
v
corresponds to the standard construction of random recursive trees (RRTs). These are
6
5 constructed by sequentially adding new vertices, which are attached to a uniformly random
6 vertex in the previous tree. Second, a class of ‘static’ trees formed via a coalescent process
1 beginning with n isolated nodes. They described the construction of a ‘static’ tree with n
0
vertices as follows.
.
1
“Initially, single elements move about at random. Each collision forms a
0
couple. Acollisionofacouplewithasingleelementformsatriple,acollision
7
1 of an s-tuple with a t-tuple forms an (s + t)-tuple, and so on. At each
: collision a link is established between an element of one X-tuple and an
v
element of another, the links being rigid so that the elements of the same
i
X
k-tuplecannotcollide. Theprocessgoesonuntiltheentiresetofnelements
r has been joined into an n-tuple.”
a
The term ‘static’ was motivated by the fact that this construction starts with the n
vertices the tree is aimed to have at the end of the process. This is a description, in
fact, of a discrete multiplicative coalescent which is linked to Kruskal’s algorithm for the
minimum weighted spanning tree problem [1] 1. A growing process of such coalescent was
not foreseen; however, it is now known that, for some coalescent procedures (e.g. additive
Date:January6th2017.
2010 Mathematics Subject Classification. 60C05,05C80.
Key words and phrases. Kingman’s coalescent, random recursive trees, extreme values, tree growth
processes,Chen-Steinmethod,coupling.
1Unfortunately,itwasincorrectlypresumedin[16]tobuilduniformlyrandomunrootedlabeledtrees.
1
2 LAURAESLAVA
and Kingman’s), the resulting tree can also be constructed by a growth process [1, 15, 17].
In particular, Kingman’s coalescents correspond to RRTs; see e.g. [1], or Propositions 1.1
and 5.3 below.
The key conceptual contribution of this work is what we call the Robin-Hood pruning
procedure. Thisisarandomconstructionwhich,givenaKingman’scoalescentonnvertices,
produces a Kingman’s coalescent on n+1 vertices. The benefits of such construction are
twofolded.
First,Kingman’scoalescenthadalreadybeenexploitedbyAddario-Berryandtheauthor
to describe near-maximum degrees in RRTs, [2, 7]. With the new procedure, we are able to
extract finer information about extreme degree values in RRTs. Second, growth procedures
naturally couple families of trees as the size varies. However, typically there is no simple
coupling of finite n-coalescent processes as n varies. The introduction of the Robin-Hood
pruning provides, to the best of our knowledge, a novel tree growth procedure which is
interesting on its own; thereby, opening a wide range of further avenues of research.
TheRobin-Hoodpruningisbestdescribedthroughanauxiliarytreestructurethatrelates
to both Kingman’s coalescent and RRTs. We proceed to its description, then we present
the results obtained in this work.
1.1. Notation. Wedenotenaturallogarithmsbyln(·)andlogarithmswithbase2bylog(·).
For n∈N, we write [n]={1,...,n} and let S be the set of permutations on [n].
n
Given a rooted labeled tree t = (V(t),E(t)), write |t| = |V(t)| and call |t| the size of t.
We write T for the set of rooted trees t with vertex set V(t) = [n]. By convention, we
n
direct all edges toward the root r(t) and write e = uv for an edge with tail u and head v.
For u∈V(t)\{r(t)} we write p (u) for the parent of u, that is, the unique vertex v with uv
t
in E(t). Finally, write d (v) for the number of edges directed toward v in t, and call d (v)
t t
the degree of v. Note that d (v)=#{u:p (u)=v}.
t t
We say t∈T is increasing if its vertex labels increase along root-to-leaf paths; in other
n
words, if t ∈ T and p (v) < v for all v ∈ [n]\{r(t)} (in particular, r(t) = 1). We write
n t
I ⊂T for the set of increasing trees of size n. It is easy to see that |I |=(n−1)! for all
n n n
n. Next, a tree growth process is a sequence (t , n ≥ 1) of trees with t ∈ T for each n.
n n n
The process is increasing if t is a subtree of t for all n; this implies that t ∈I for all
n n+1 n n
n.
TreegrowthprocessesselectatreefromT foreachn≥1usuallywiththecharacteristic
n
that new vertices attach to some vertex in the previous tree, giving rise to increasing trees.
1.2. Recursively decorated trees. We begin with the standard construction of a RRT
of size n ≥ 1, which we denote by R . Start with R as a single node with label 1. For
n 1
each 1 < j ≤ n, R is obtained from R by adding a new vertex j and connecting it to
j j−1
v ∈[j−1]; the choice of v is uniformly random and independent for each 1<j ≤n. The
j j
process(R , n≥1)isarandomincreasingtreegrowthprocess. Moreover, itisreadilyseen
n
that R is a random increasing tree uniformly chosen from I .
n n
Recursively decorated trees extend the concept of increasing trees. If t∈T and σ ∈S
n n
then σ(t) is the tree t(cid:48) ∈ T with edges {σ(u)σ(v) : uv ∈ E(t)}. We say σ is an addition
n
history for t if σ(t) is increasing. If σ is an addition history for t then we say that the pair
(t,σ) is a recursively decorated tree or decorated tree, and that vertex v has addition time
σ(v), for all v ∈V(t). Write
RD ={(t,σ): t∈T , σ is an addition history of t},
n n
for the set of recursively decorated trees of size n. See Figure 1 for an example.
EXTREMAL VALUES IN RRTS 3
1
2 1
2 3 6
6 3 5 2 3 6
4 5
4 1 4 5
(t,σ) σ(t)
Figure 1. A decorated tree (t,σ) ∈ RD on the left; the permutation σ
6
is depicted with bold numbers next to the vertices in t (so for example
σ(1)=5 and σ(6)=2). On the right, the increasing tree σ(t).
For each n ≥ 1 let RT = (T ,σ ) be a uniformly chosen decorated tree in RD . We
n n n n
remarknowthatσ encodestheevolutionofKingman’scoalescentonnvertices;thedetails
n
onsuchcorrespondancearegiveninSection5. Thenext,straightforwardpropositionshows
that the shape of T has the same law as that of R .
n n
Proposition 1.1. For each n ∈ N, |RD | = n!(n−1)! and if RT = (T ,σ ) ∈ RD is
n n n n n
chosen uniformly at random then σ (T ) is a random recursive tree of size n and σ is a
n n n
uniformly chosen permutation in S .
n
Proof. Bydefinition,if(t,σ)∈RD ,thenσ(t)∈I . Letϕ:RD →I ×S bedefinedsuch
n n n n n
that ϕ(t,σ)=(σ(t),σ). For an increasing tree t and σ ∈S , let t(cid:48) =σ−1(t) then ϕ(t(cid:48),σ)=
n
(t,σ),itisalsostraightforwardthatϕisinjective. Thus|RD |=|I |·|S |=n!(n−1)!. The
n n n
resultfollowssincebijectionspreservetheuniformmeasureonfiniteprobabilityspaces. (cid:3)
Corollary 1.2. For all n∈N, the following distributional identity holds.
(d (σ−1(v)); v ∈[n])d=ist(d (v); v ∈[n]).
RTn n Rn
1.3. Statement of results. The Robin-Hood pruning RH :RD →RD is a random
n n−1 n
procedure that allows us to construct RT from RT while preserving most of the edges
n n−1
in RT .
n−1
Broadly speaking, RH (t,σ) is obtained from (t,σ) by pruning some subtrees of t and
n
placing them as subtrees of a new vertex labeled n; additionally, vertex n attaches to a
random vertex or becomes the root of the new tree. The addition history in RH (t,σ) is
n
adjustedfromσ suchthatvertexnhasauniformlyrandomaddition time. Heuristically,the
random procedure follows a ‘steal from the old to give to the new’ scheme; that is, once the
addition time of n has been determined, vertices which had been added earlier (according
to σ) have larger probability of being reattached to vertex n.
We will write RH=RH when the size of the input is clear from the context. The exact
n
definition of RH will be given in the next section along with the proof of the following
n
result.
Theorem 1.3. For each n≥1 let RT =(T ,σ ) be a uniformly random element in RD .
n n n n
The Robin-Hood pruning provides a coupling for ((T ,σ ), n ≥ 1) by setting (T ,σ ) =
n n n n
RH(T ,σ ) for each n≥2.
n−1 n−1
4 LAURAESLAVA
As we will establish in Proposition 5.3, RT is a representation of Kingman’s coalescent
n
on [n]. Therefore, we have the following corollary.
Corollary 1.4. The construction of ((T ,σ ), n ≥ 1) in Theorem 1.3 gives an explicit
n n
coupling of all finite Kingman’s coalescents.
A remarkable property of the coupling in Theorem 1.3 is that, it yields a tree growth
processwhereallthetreesaredistributedasRRTs,howevertheprocessitselfisnotincreas-
ing. To the best of our knowledge, this is a novel evolution of random networks. Potential
applications and open problems are discussed in Section 6.
Turning to extreme values in the degree sequence of RRTs, consider the following vari-
ables. For integers 0<m≤n, let
X(n) =#{v ∈[n]:d (v)=m};
m Rn
Janson established the joint limiting distribution of (X(n), m ≥ 1) in [12]; for previous
m
results on the degree distribution of RRTs see the survey [18]. In terms of capturing the
degree sequence around the maximum degree range, Addario-Berry and the author provide
all the possible limiting distributions of (X(n) , k ∈Z) in [2].
(cid:98)logn(cid:99)+k
In this work we are concerned with high-degree vertices in a broader sense; that is, we
consider the variables
Z(n) =#{v ∈[n]:d (v)≥m},
m Rn
withm∼clnnandprovideresultsonconvergenceratestowardtheirlimitingdistributions.
(cid:104) (cid:105)
(n)
Throughout the paper, we write λ =E Z and record the following estimate.
n,m m
Lemma 1.5 (Lemma 4.3 in [2]). First, λ ≤2−m+logn, and for each c∈(0,2), there is
n,m
γ(c) such that, uniformly over m<clnn,
λ =nP(d (1)≥m)=2−m+logn(1+o(n−γ)).
n,m RTn
Our first result on high-degree vertices is obtained by applying the Chen-Stein method.
Theorem 1.6. Fix1<c<c(cid:48) <2. Thereareconstantsα=α(c(cid:48))∈(0,1)andβ =β(c)>0
such that uniformly for m=m(n) satisfying clnn<m<c(cid:48)lnn,
(cid:16) (cid:17)
d Z(n),Poi(λ ) ≤O(2−m+(1−α)logn)+O(n−β).
TV m n,m
The exponent −m+(1−α)logn in Theorem 1.6 is negative when (1−α)loge < c. A
detailed but simple track of the conditions on α, see Proposition 3.2, shows that there is
a non-empty interval Ic(cid:48) = ((1−α)loge,c(cid:48)) such that if c ∈ Ic(cid:48) ∩(1,2), then the bounds
in Theorem 1.6 are, in fact, tending to zero. Moreover, by Lemma 1.5, if c < loge, then
λ →∞ as n→∞. This fact, together with Theorem 1.6 yields the next corollary.
n,clnn
Corollary 1.7. For each c(cid:48) ∈ (1,loge) there exists c ∈ (1,c(cid:48)) such that if clnn < m <
c(cid:48)lnn, then
Zm(n)−λn,m −di→st N(0,1).
(cid:112)
λ
n,m
Corollary 1.7 extends the range of m=m(n) for which a central limit theorem exists for
Z(n); previous results were given for m constant [12] and for m = logn−d with d = d(n)
m
slowly tending to infinity [2].
EXTREMAL VALUES IN RRTS 5
We opted to state Theorem 1.6 and Corollary 1.7 separately to clarify where the bounds
on m are limiting the convergence rates. In Section 3, we explain how previous results in
[2]determinetheexponentα, whiletheexponent β dependsonanauxiliarycouplingbased
on the Robin-Hood pruning. The details of such coupling are given in Section 4; for the
moment we remark that, by Corollary 1.2, for all m≤n,
(1) Z(n) d=ist#{v ∈[n]:d (v)≥m};
m RTn
when there is no ambiguity, we write RT to refer only to its tree coordinate. The pruning
n
procedure provides a key description of (d (i), i ∈ [n]) in terms of both (d (i), i ∈
RTn RTn−1
[n−1]) and d (n), which is independent of the former vector. This allows us to analyze
RTn
the conditional law of (d (i), i∈[n−1]) given that d (n)≥m.
RTn RTn
Our last result concerns the maximum degree ∆ of R , which by Corollary 1.2 satisfies
n n
dist
∆ = max{d (v): v ∈[n]}.
n RTn
Theorem 1.8. There exists C >0 such that uniformly over 0<i=i(n)<logelnlnn−C,
P(∆ <(cid:98)logn(cid:99)−i)=exp{−2i+εn}(1+o(1)),
n
where ε =logn−(cid:98)logn(cid:99).
n
The first bounds of this type, for P(∆ <(cid:98)logn(cid:99)+j) with j ∈Z, were given in [9]. An
n
extension to j < 2lnn−logn was obtained in [2]. Goh and Schmutz provide a heuristic
of how the Gumbel, or double-exponential, distribution arises for ∆ [9]. They do so by
n
lookingatthelimitingdistributionofd (i)withi→∞slowly. Belowwepresentadistinct
Rn
heuristic in terms of the degree distribution of RT .
n
Themaximumofi.i.d. randomvariablesis,underrathergeneralconditions,distributedin
thelimitastheGumbeldistribution[11]. Latticedistributionsareexcludedfromthisregime
and, instead, Anderson gives analogous conditions under which the Gumbel distribution
servesasanapproximationfortheirmaximum[3]. Inthecaseof(deg (v), v ∈[n]), their
RTn
limiting distributions are geometric, a distribution which satisfies the conditions given in
[3]. Although,thedegreesofatreearenotindependent,theircorrelationsareweakandthe
Gumbel-type approximation still arises for the distribution of ∆ .
n
Outline. The remainder of the paper is organized as follows. The proof of Theorem 1.3
is given in the next section, and its connection to Kignman’s coalescent in Section 5. In
(n)
Section 3, we explain how we apply the Chen-Stein method to Z by constructing an
m
auxiliary coupling; the proofs of Theorems 1.6 and 1.8 and Corollary 1.7 are provided in
Section 3 under the assumption of the existence of such coupling. The auxiliary coupling is
basedontheRobin-HoodpruningandisdefinedinSection4. Toclosethiswork, webriefly
discuss further avenues of research in Section 6.
2. The Robin-Hood pruning
For each n ≥ 2, the Robin-Hood pruning RH is a random procedure that takes a
n
decorated tree (t,σ) ∈ RD and outputs a decorated tree RH (t,σ) ∈ RD . We first
n−1 n n
define a deterministic pruning, which is illustrated in Figure 2.
2.1. A deterministic process. For d≥1, we write x=(x ,...,x )∈{0,1}d. Let n>1
1 d
and set
C ={(k,l,x): 1≤l<k ≤n, x∈{0,1}n−1}∪{(1,0,x): x∈{0,1}n−1, x =1};
n 1
6 LAURAESLAVA
additionally, for (k,l,x)∈C and a permutation σ ∈S , let
n n−1
V (k,l,x,σ)=V (k,x,σ)={v ∈[n−1]: x =1, σ(v)≥k}.
n n σ(v)
Note that the definition of C is such that σ−1(1) ∈ V if and only if k = 1. The set V
n n n
correspondstotheverticestobeprunedandrewiredinthefollowingdeterministicpruning.
Definition 2.1. For n ≥ 2, (t,σ) ∈ RD and (k,l,x) ∈ C define (t(cid:48),σ(cid:48)) ∈ T ×S as
n−1 n n n
follows.
First, t(cid:48) is obtained from t as follows. Let V = V (k,x,σ). For each v ∈ V \{r(t)},
n
replace the edge vp (v) with an edge connecting v to a new vertex labeled n. Now, if k = 1
t
then attach r(t) to n; otherwise, attach vertex n to σ−1(l). In other words, the edges of t(cid:48)
are given by
(cid:40)
(E(t)∪{vn; v ∈V})\{vp (v); v ∈V} if k =1,
E(t(cid:48))= t
{nσ−1(l)}∪(E(t)∪{vn; v ∈V})\{vp (v); v ∈V} if k >1.
t
Second, let σ(cid:48) :[n]→[n] be defined by σ(cid:48)(n)=k and for v <n,
σ(cid:48)(v)=σ(v)+1 .
[σ(v)≥k]
We write rh ((t,σ),(k,l,x))=(t(cid:48),σ(cid:48)).
n
Lemma 2.2. Fix n≥2. For any (t,σ)∈RD and (k,l,x)∈C ,
n−1 n
rh ((t,σ),(k,l,x))∈RD .
n n
Proof. Write rh ((t,σ),(k,l,x)) = (t(cid:48)σ(cid:48)). When k = 1, it is clear that t(cid:48) is a tree. When
n
k > 1, let w = σ−1(l) be the parent of n in t(cid:48) and let (w = v ,...,v = r(t)) be the path
1 j
from w to the root of t. Since σ is an addition history of t,
l=σ(v )>σ(v )>···>σ(v )=1;
1 2 j
moreover, l < k. It follows that v ∈/ V(k,l,x,σ) for i ∈ [j] and consequently, no edges in
i
the path from n to the root in t(cid:48) closes a cycle by connecting to n.
Now, we show that σ(cid:48) is an addition history for t(cid:48). It is clear that σ(cid:48) is a permutation of
[n],soitsufficestoprovethatσ(cid:48)(v)>σ(cid:48)(pt(cid:48)(v)),forallv ∈V(t)\{r(t(cid:48))}. First,forvertices
v with pt(cid:48)(v)=n we have σ(v)≥k and consequently
σ(cid:48)(v)=σ(v)+1>k =σ(cid:48)(n).
Second, considerv,w <nwithpt(cid:48)(v)=w. Itfollowsthatvw ∈E(t)andthusσ(v)>σ(w).
Consequently, 1 ≥1 and so σ(cid:48)(v)>σ(cid:48)(w). The last case occurs when k >1
[σ(v)≥k] [σ(w)≥k]
and pt(cid:48)(n)=w =σ−1(l). We then have
σ(cid:48)(n)=k >l=σ(w)=σ(cid:48)(w). (cid:3)
We note here a property of this pruning procedure that will be useful in the proof of
Proposition 1.6; or more precisely, Proposition 4.3.
Fact 2.3. Fix n≥2. For any (t,σ)∈RD and (k,l,x)∈C , write
n−1 n
(t(cid:48),σ(cid:48))=rh ((t,σ),(k,l,x))∈RD .
n n
Then dt(cid:48)(n)=(cid:80)ni=−k1xi, and for v ∈[n−1],
n−1
(cid:88)
dt(cid:48)(v)=dt(v)+1[l=σ(v)]− xi1[v=pt(σ−1(i))].
i=k
EXTREMAL VALUES IN RRTS 7
1
3
2 4 5
7 5 2
3 7 6 8 9
8 4 1 9 6
(a) A tree (t,σ) in RD . The permutation σ is depicted with bold numbers.
9
1
3
2 4 5 l=5
7 5 2
3 7 6 8 9 k=6
8 4 1 9 6 10
(b) In this case, k=6,l=5 and x =x =x =x =1; all other x =0. Vertices in gray satisfy
1 2 7 8 i
X =1 and underlined are addition times σ(i)≥k.
σ(v)
1
3
2 4 5 5
7 5 2 10 10
3 7 10 6 8 9
8 1 6 10 4 9
(c) Nodes i with σ(i)≥k and x =1 have been pruned and addition times have been adjusted.
i
1
3
2 4 5
7 5 2
3 7 10 6
8 1 6 10
8 9
4 9
(d) The resulting tree rh ((t,σ),(k,l,x))∈RD .
10 10
Figure 2. An example of the Robin-Hood pruning for n=10.
8 LAURAESLAVA
2.2. The random process. The Robin-Hood pruning is defined, for each (t,σ) ∈ RD ,
n
by
RH (t,σ)=rh ((t,σ),(K,L,X));
n n
where the element (K,L,X)∈C is an RH set of random variables defined as follows.
n n
dist
Definition2.4. Fixn≥1. LetK = Unif(1,2,...,n); ifK =1letL=0,andifK >1let
L = Unif(1,2,...K−1). Independently, let X = Bernoulli(1/i) be independent variables
i
for i ∈ [n−1] and write X = (X ,...,X ). An RH -set is a triple of random variables
1 n−1 n
with the same law as (K,L,X).
The law of RH (t,σ) depends on the initial input (t,σ); however, the distribution of
n
the RH -set of variables is defined so that RH (RT ) preserves the uniform measure in
n n n−1
decorated trees. To verify this claim, we require the following characterization of RT .
n
Lemma 2.5. Let n≥1 be an integer. A random decorated tree (T,σ)∈RD is uniformly
n
random if and only if the following properties are satisfied.
i) The permutation σ is uniformly random on S .
n
ii) The vertices
(p (v), v ∈V(T)\{r(T)})=(p (σ−1(v)), v ∈V(T)\{r(T)})
T σ(T)
are, conditionally given σ, independent.
iii) For all vertices v,w ∈[n] and indices i,j ∈[n],
1
(2) P(p (v)=w, σ(v)=j, σ(w)=i)= 1 .
T n(n−1)(j−1) [j>i]
Proof. Let (T,σ) be uniformly random on RD . Then by Proposition 1.1, σ is a uniformly
n
random permutation and σ(T) has the law of R . Therefore, as multisets,
n
dist
{p (v), v ∈V(T)\{r(T)}} = {p (v), 1<v ≤n};
T σ(T)
and parents in RRTs are chosen independently for each of the vertices. The third condition
follows immediately: For all v,w,i,j ∈[n], we obtain
1
P(p (v)=w, σ(v)=j, σ(w)=i)= P(p (v)=w|σ(v)=j, σ(w)=i)
T n(n−1) T
1 (cid:0) (cid:1)
= P p (j)=i
n(n−1) σ(T)
1
= 1 .
n(n−1)(j−1) [j>i]
Now consider a random decorated tree (T,σ)∈RD satisfying conditions i)-iii). Fix a
n
decorated tree (t,π)∈RD , and for v ∈V(t)\{r(t)}, let w =p (v), then
n v t
P(p (v)=w |σ =π)=P(p (v)=w |σ(v)=π(v), σ(w )=π(w ))
T v T v v v
1
(3) = .
π(w )−1
v
The first equality holds by condition ii) and the second by both i) and iii) since π(v) >
π(w ).
v
EXTREMAL VALUES IN RRTS 9
Now, by definition, π(t)∈I . The increasing tree t(cid:48) is determined by the set of parents
n
{pt(cid:48)(v), 1<v ≤n}. Therefore,
(cid:0) (cid:1)
P(σ(T)=π(t)|σ =π)= P p (v)=p (v), 1<v ≤n|σ =π
σ(T) π(t)
= P(p (v)=p (v), v ∈V(t)\{r(t)}|σ =π)
T t
(cid:89)
= P(p (v)=p (v)|σ =π)
T t
v∈V(t)\r(T)
= [(n−1)!]−1.
Condition ii) gives the third equality; the last equality holds by (3) since
{π(w ), v ∈V(t)\{r(t)}}={2,...,n}.
v
Finally condition i) and the computations above show that, regardless of the choice of
(t,π)∈RD , we have
n
P((T,σ)=(t,π))=P(σ(T)=π(t)|σ =π)P(σ =π)
1
= P(σ(T)=π(t)|σ =π)=[n!(n−1)!]−1.
(cid:3)
n!
We are now ready to prove Theorem 1.3.
Proof of Theorem 1.3. Let (T,σ) ∈ RD be a uniformly random decorated tree. Let
n−1
(K,L,X) be an RH set and let (T(cid:48),π) = rh((T,σ),(K,L,X)). It suffices to show that
n
(T(cid:48),π) satisfies the properties in Lemma 2.5.
First,conditioni)followsfromtheconstructionofπ andthedistributionsofbothK and
σ. Second, once conditioning on π, which is equivalent to conditioning on both σ and K,
we get
{pT(cid:48)(v), v ∈V(T(cid:48))\{r(T(cid:48))}}={pT(cid:48)(v), 1<π(v)<π(n)}
∪{pT(cid:48)(v), π(n)≤π(v)≤n}
={p (v), v ∈1<σ(v)<K}
T
∪{pT(cid:48)(v), (2∨K)≤π(v)≤n},
wherethelasttwosetsareconditionallyindependentgivenπ. Now,since(T,σ)isuniformly
random in RD , the parents {p (v), v ∈ 1 < σ(v) < K} are independent, conditionally
n−1 T
given σ (and thus, also conditionally given π). On the other hand, for v with π(v)≥K,

n if X =1,
 π(v)−1
pT(cid:48)(v)= pT(v) if Xπ(v) =0,
π−1(L) if π(v)=K.
NotethatpT(cid:48)(v)isdeterminedindependentlyfromothervertices,thus{pT(cid:48)(v), K ≤π(v)≤
n} are also independent, conditionally given π. This implies that condition ii) is satisfied.
Third, fix 1 ≤ i < j ≤ n and fix distinct v,w ∈ [n]. We consider three cases; namely
v =n, w =n, and v,w ∈[n−1]. Let
A1 ={pT(cid:48)(n)=w, π(n)=j, π(w)=i},
A2 ={pT(cid:48)(v)=n, π(v)=j, π(n)=i},
A3 ={pT(cid:48)(v)=w, π(v)=j, π(w)=i}.
10 LAURAESLAVA
ItremainstoshowthattheprobabilitiesofA ,A ,A aregivenby (2)foralli,j ∈[n]. The
1 2 3
event pT(cid:48)(n)=w implies that σ(w)=L<K. Therefore, A1 occurs precisely when K =j,
L=i, and σ(w)=i. Then,
1
P(A )=P(K =j,L=i)P(σ(w)=i)= .
1 n(j−1)(n−1)
Next, pT(cid:48)(v)=n implies that σ(v)≥K and thus π(v)=σ(v)+1. It then follows that A2
occurs when K =i, σ(v)=j−1, and X =1. Therefore,
j−1
1
P(A )=P(K =i,X =1)P(σ(v)=j−1)= .
2 j−1 n(j−1)(n−1)
For the last case, since u,v <n, it follows that K ∈/ {i,j}. For each k ∈[n]\{i,j} let
A3,k ={pT(cid:48)(v)=w, π(v)=j, π(w)=i, K =k}.
IncomputingtheprobabilitiesP(A )weusethat(T,σ)isuniformlyrandominRD . If
3,k n−1
K >j,thenbothσ(v)=π(v)andσ(w)=π(w);inaddition,pT(cid:48)(v)=wonlyifpT(v)=w.
Therefore, if k >j, then
P(A )=P(K =k)P(p (v)=w, σ(v)=j, σ(w)=i)
3,k T
1
= .
n(n−1)(n−2)(j−1)
Similarly,ifK <j,thenσ(v)=π(v)−1,σ(w)=π(w)−1 ,andadditionallyX =0.
[K<i] j−1
It then follows that, if k <j,
(cid:0) (cid:1)
P(A )=P(K =k, X =0)P p (v)=w, σ(v)=j−1, σ(w)=i−1
3,k j−1 T [K<i]
1 j−2 1
= · · .
n j−1 (n−1)(n−2)(j−2)
We have shown that P(A ) is uniform for all k ∈[n]\{i,j}, and we get
3,k
(cid:88) 1
P(A )= P(A )= .
3 3,k n(n−1)(j−1)
k(cid:54)=i,j
Altogether, we have shown that condition iii) is satisfied and so the proof is complete. (cid:3)
3. Large degrees in RRTs
The aim in this section is to bound the convergence rate of the law of 2
Z(n) d=ist#{v ∈[n]:d (v)≥m}
m RTn
to a suitable Poisson random variable. Our tool in this section is the Chen-Stein method as
stated in Proposition 3.1 below. Given probability measures µ and ν, a coupling of µ and ν
is a pair (X,Y) of random variables (either real or vector-valued) with X ∼µ and Y ∼ν.
Let I = (I , a ∈ A) be a collection of {0,1}-valued random variables. Let µ be the law
a
(cid:80)
of W = I and for a∈A let ν be the conditional law of W given that I =1, so
a∈A a a a
ν (B)=P(W ∈B|I =1).
a a
2ByFact1.2,consideringeitherRn orRTn inthedefinitionofZm(n) isequivalent.