We consider the computer generation of random binary search trees with n nodes for the standard random permutation model . The algorithms discussed here output the number of external nodes at each level, but not the shape of the tree. This is important, for example, when one wishes to simulate the height of the binary search tree . Various paradigms are proposed, including depth-first search wi...

We study a class of binary trees that grow in a random environment, where the state of the environment can change at every vertex of the trees. The trees considered are single-type and two-type binary trees that grow in a two-state Markovian environment. For each kind of tree, the conditions on the environment process for extinction of the tree are determined, and the problem of calculating the...

We consider the universal Grigorchuk 2-group, i.e., the group such that every Grigorchuk 2-group is a quotient. We show that this group has a nice universal representation in the group of all functions Func({0, 1, 2}N, Aut(T2)), where T2 is a group of automorphism of the binary tree. Finally, we prove that this universal Grigorchuk 2-group is amenable. The proof is an application of the “Munchh...

Rotation distance quantifies the difference in shape between two rooted binary trees of the same size by counting the minimum number of elementary changes needed to transform one tree to the other. We describe several types of rotation distance, and provide upper bounds on distances between trees with a fixed number of nodes with respect to each type. These bounds are obtained by relating each ...

A b s t r a c t

Binary trees can be made differentially private by adding noise to every node and leaf. In such form they allow multifaceted exploration of a variable without revealing any individual information. While a differentially private binary tree can be used and read just like its conventional exact-valued analog, realizing that different combinations of nodes contain overlapping answers to the same i...

A class of binary trees that grow in a random environment,where the state of the environment can change at every vertex of thetrees is studied. The trees considered are single-type and two-type binarytrees that grow in a two-state Markovian environment.For each kind oftree, the conditions on the environment process for extinction of the treeare determined, and the proble...

We present a combinatorial proof of Postnikov’s hook length formula for binary trees. AMS Classification: 05A15, 05A19

A widely-studied model for generating binary sequences is to ‘evolve’ them on a tree according to a symmetric Markov process. We show that under this model distinguishing the true (model) tree from a false one is substantially “easier” (in terms of the sequence length needed) than determining the true tree. The key tool is a new and tight Ramsey-type result for binary trees.

The rotation distance d(S, T ) between two binary trees S, T of n vertices is the minimum number of rotations to transform S into T . While it is known that d(S, T ) ≤ 2n− 6, a well known conjecture states that there are trees for which this bound is sharp for any value of n ≥ 11. We are unable to prove the conjecture, but we give here some simple criteria for lower bound evaluation, leading fo...