12 problems

Let $w$ be a Dyck path and let $t=\kappa(w)$ be the corresponding planar rooted binary tree under the bijection $\kappa$ from Dyck paths to planar rooted binary trees. A correspond…

First signed path conjecture. For every pair of finite, binary trees $(D,R)$ with the same number of leaves, there is a sign assignment of $D$ and a word $w$ of rotation symbols va…

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

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

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.

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…

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 $\mathcal{A}_{n-2}^{s}$.

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.