21 problems

Let $\mathcal D_n$ be the disordered random full binary tree with $n$ leaves, and let $N_t(\mathcal D_n)$ denote the number of occurrences of a fixed full binary tree $t$ as a frin…

Peripheral embedding conjecture. $G$ quasiisometrically embeds into a product of binary trees if and only if each peripheral subgroup $H_t$ does as well.

Let the sum range over all unlabeled complete binary trees $T$ with $n$ internal vertices, let the product range over all internal vertices $v$ of $T$, and let $h_v$ be the number…

Pinned-spine conjecture. Every FBT $T$ on $n$ vertices admits a graceful labeling $f$ such that, for some longest root-to-leaf path $v_0v_1\cdots v_\ell$, the restriction of $f$ to…

Equality conjecture. For all relevant $n$, $p$, and $q$, one has

Let $\ell$ be a positive integer, and consider stable configurations of $2^{\ell}-1$ labeled chips on a binary tree. For a vertex $v$ and an integer $i$, compare the $i$th smallest…

Ballot-property conjecture. In a stable configuration, the whole tree and every subtree satisfy the ballot property.

Consider a binary tree with arbitrary positive edge weights $(W_e)_{e\in E}$ that are not necessarily determined by the depth of the edge. For $p\neq 2$, let a linear extension ope…

Most likely terminal configuration conjecture. For all $n$, the most likely terminal configuration is the unique binary search tree on the complete binary tree with $2^n-1$ nodes.

Paired-fractal conjecture. Every such equivalent or formally identical system will exhibit both the Takagi curve and the $\beta$-Cantor function.

Czabarka–et al. conjecture. For every binary tree $B$, the asymptotic formula

Let $B$ be a binary tree, let $\gamma(B,T)$ denote the induced density of copies of $B$ in a binary tree $T$, and let $|T|$ denote the order of $T$. Let $i(B)$ be the inducibility…

Let $\operatorname{E}_n$ denote the rooted binary tree with $n$ leaves, and let $c(F,T)$ be the number of copies of a rooted binary tree $F$ in a rooted binary tree $T$. For intege…

Consider the symmetric discrete-time quantum walk on the semi-infinite binary tree and the corresponding classical symmetric random walk, with $n$ denoting the initial level in the…

Let $n \geq 4$, and let $T_1$ and $T_2$ be $n$-leaf path trees such that leaf $1$ is on level $1$ in $T_1$ and leaf $n$ is on level $1$ in $T_2$. A pair is mutually crooked if it c…

Let $T_1'$ and $T_2'$ be $(n-1)$-leaf binary trees. Let $1 \leq i \leq n-1$, and let $T_1$ and $T_2$ be the $n$-leaf trees obtained from $T_1'$ and $T_2'$ by duplicating leaf $i$.…

Eliahou–Kryuchkov conjecture. For any pair of vertices on $\mathcal{A}_{n-2}$ there exists a path connecting them that can be lifted to a path on the graph $\mathcal{A}_{n-2}^{s}$.

Let $n=2p-1$ or $n=2p$ and define the binary trees … and … Here $\operatorname{dist}(T,T')$ denotes the rotation distance between binary trees $T$ and $T'$. The maximal rotation di…

Replacement-bijection conjecture. If $s$ and $t$ are equivalent, then there is a sequence of top-down replacements, bottom-up replacements, and left--right reflections that produce…

Avoiding-enumerating equivalence conjecture. If $s$ and $t$ are avoiding-equivalent, then they are also enumerating-equivalent.

Optimal transition-probability conjecture. The optimal transition probabilities are