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 10 such that uniformly for m=m(n) satisfying clnn0 such that uniformly over 01 1 d and set C ={(k,l,x): 1≤l1. t Second, let σ(cid:48) :[n]→[n] be defined by σ(cid:48)(n)=k and for v 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 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), 1i] 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), 1j,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

Extremal values in recursive trees via a new tree growth process

Similar Extremal values in recursive trees via a new tree growth process