On the inducibility of small trees (original) (raw)
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Inducibility of topological trees
Trees without vertices of degree 2 are sometimes named topological trees. In this work, we bring forward the study of the inducibility of (rooted) topological trees with a given number of leaves. The inducibility of a topological tree S is the limit superior of the proportion of all subsets of leaves of T that induce a copy of S as the size of T grows to infinity. In particular, this relaxes the degree-restriction for the existing notion of the inducibility in d-ary trees. We discuss some of the properties of this generalised concept and investigate its connection with the degree-restricted inducibility. In addition, we prove that stars and binary caterpillars are the only topological trees that have an inducibility of 1. We also find an explicit lower bound on the limit inferior of the proportion of all subsets of leaves of T that induce either a star or a binary caterpillar as the size of T tends to infinity.
arXiv (Cornell University), 2018
Imitating a recently introduced invariant of trees, we initiate the study of the inducibility of d-ary trees (rooted trees whose vertex outdegrees are bounded from above by d ≥ 2) with a given number of leaves. We determine the exact inducibility for stars and binary caterpillars. For T in the family of strictly d-ary trees (every vertex has 0 or d children), we prove that the difference between the maximum density of a d-ary tree D in T and the inducibility of D is of order O(|T | −1/2) compared to the general case where it is shown that the difference is O(|T | −1) which, in particular, responds positively to an existing conjecture on the inducibility in binary trees. We also discover that the inducibility of a binary tree in d-ary trees is independent of d. Furthermore, we establish a general lower bound on the inducibility and also provide a bound for some special trees. Moreover, we find that the maximum inducibility is attained for binary caterpillars for every d.
Further results on the inducibility of ddd-ary trees
arXiv (Cornell University), 2018
A subset of leaves of a rooted tree induces a new tree in a natural way. The density of a tree D inside a larger tree T is the proportion of such leaf-induced subtrees in T that are isomorphic to D among all those with the same number of leaves as D. The inducibility of D measures how large this density can be as the size of T tends to infinity. In this paper, we explicitly determine the inducibility in some previously unknown cases and find general upper and lower bounds, in particular in the case where D is balanced, i.e., when its branches have at least almost the same size. Moreover, we prove a result on the speed of convergence of the maximum density of D in strictly d-ary trees T (trees where every internal vertex has precisely d children) of a given size n to the inducibility as n → ∞, which supports an open conjecture.
On the inducibility of rooted trees
2018
On the Inducibility of Rooted Trees A. A. V. Dossou-Olory Department of Mathematical Sciences Mathematics Division University of Stellenbosch Private Bag X1, Matieland 7602, South Africa. Dissertation: PhD (Mathematics)
Inducibility and universality for trees
Combinatorial Theory
We answer three questions posed by Bubeck and Linial on the limit densities of subtrees in trees. We prove there exist positive ε 1 and ε 2 such that every tree that is neither a path nor a star has inducibility at most 1 − ε 1 , where the inducibility of a tree T is defined as the maximum limit density of T , and that there are infinitely many trees with inducibility at least ε 2. Finally, we construct a universal sequence of trees; that is, a sequence in which the limit density of any tree is positive.
On isomorphism classes of leaf-induced subtrees in topological trees
2022
A subtree can be induced in a natural way by a subset of leaves of a rooted tree. We study the number of nonisomorphic such subtrees induced by leaves (leaf-induced subtrees) of a rooted tree with no vertex of outdegree 1 (topological tree). We show that only stars and binary caterpillars have the minimum nonisomorphic leaf-induced subtrees among all topological trees with a given number of leaves. We obtain a closed formula and a recursive formula for the families of d-ary caterpillars and complete d-ary trees, respectively. An asymptotic formula is found for complete d-ary trees using polynomial recurrences. We also show that the complete binary tree of height h > 1 contains precisely 2(1.24602...) 2 h nonisomorphic leaf-induced subtrees.
Note on the Exponential Recursive k-Ary Trees
Theoretical Informatics and Applications, 2023
In the present paper, we consider exponential recursive trees with no node of outdegree greater than k, called exponential recursive k-ary trees (k ≥ 2). At each step of growing of these trees, every external node (insertion position) is changed into a leaf with probability p, or fails to do so with probability 1 − p. We investigate limiting behavior of fundamental parameters such as size, leaves and distances in exponential recursive k-ary trees.
Discrete Mathematics, 2011
We prove a finitary version of the Halpern-Läuchli Theorem. We also prove partition results about strong subtrees. Both results give estimates on the height of trees.
On the Tree Packing Conjecture
SIAM Journal on Discrete Mathematics, 2013
The Gyárfás tree packing conjecture states that any set of n−1n-1n−1 trees T1,T2,...,Tn−1T_{1},T_{2},..., T_{n-1}T1,T2,...,Tn−1 such that TiT_iTi has n−i+1n-i+1n−i+1 vertices pack into KnK_nKn. We show that t=1/10n1/4t=1/10n^{1/4}t=1/10n1/4 trees T1,T2,...,TtT_1,T_2,..., T_tT1,T2,...,Tt such that TiT_iTi has n−i+1n-i+1n−i+1 vertices pack into Kn+1K_{n+1}Kn+1 (for nnn large enough). We also prove that any set of t=1/10n1/4t=1/10n^{1/4}t=1/10n1/4 trees T1,T2,...,TtT_1,T_2,..., T_tT1,T2,...,Tt such that no tree is a star and TiT_iTi has n−i+1n-i+1n−i+1 vertices pack into KnK_{n}Kn (for nnn large enough). Finally, we prove that t=1/4n1/3t=1/4n^{1/3}t=1/4n1/3 trees T1,T2,...,TtT_1,T_2,..., T_tT1,T2,...,Tt such that TiT_iTi has n−i+1n-i+1n−i+1 vertices pack into KnK_nKn as long as each tree has maximum degree at least 2n2/32n^{2/3}2n2/3 (for nnn large enough). One of the main tools used in the paper is the famous spanning tree embedding theorem of Komlós, Sárközy and Szemerédi.
The topological trees with extreme Matula numbers
2020
Denote by p_m the m-th prime number (p_1=2, p_2=3, p_3=5, p_4=7, ...). Let T be a rooted tree with branches T_1,T_2,...,T_r. The Matula number M(T) of T is p_M(T_1)· p_M(T_2)·...· p_M(T_r), starting with M(K_1)=1. This number was put forward half a century ago by the American mathematician David Matula. In this paper, we prove that the star (consisting of a root and leaves attached to it) and the binary caterpillar (a binary tree whose internal vertices form a path starting at the root) have the smallest and greatest Matula number, respectively, over all topological trees (rooted trees without vertices of outdegree 1) with a prescribed number of leaves – the extreme values are also derived.