The maximum width of an arbitrary rooted tree is a balance index and is defined as the number of vertices at its most abundant level. It is a well-studied parameter in computer science (e.g., Drmota, Hwang, 2005; Devroye, Hwang, 2006), where it is often simply referred to as the width of a tree. More recently, it was suggested as a tree shape statistic for phylogenetic trees by Colijn, Gardy, 2014. It can be calculated using the function maxWidth from our R package treebalance.

Definition (Colijn, Gardy, 2014): The maximum width mW(T) of a tree T \in \mathcal{T}^\ast_n with height h(T) is defined as
mW(T) := \max\limits_{i=0,\ldots,h(T)} w_T(i),
where w_T(i) is the number of vertices v \in V(T) that have depth i.
Computation time: For every tree T\in\mathcal{T}^\ast_n, the maximum width mW(T) can be computed in time O(n) (Walter, 2022, Proposition 1; see also Fischer et al., 2023, Proposition 23.16).
Recursiveness: Open problem.
Locality: The maximum width is not local (Walter, 2022, Proposition 3; see also Fischer et al., 2023, Proposition 23.17).
Maximal value and trees with maximal value on \mathcal{T}^\ast_n: An upper bound for the maximum width for trees T\in \mathcal{T}^\ast_n which is tight for all n\in\mathbb{N}_{\geq 1} as well as a characterization of the corresponding maximal trees have already been established (see Fischer et al., 2023, Theorem 23.10).
Number of trees with maximal value on \mathcal{T}^\ast_n: Open problem.
Maximal value on \mathcal{BT}^\ast_n: Partially open problem. An upper bound for the maximum width for binary trees T\in \mathcal{BT}^\ast_n which is tight for all n\in\mathbb{N}_{\geq 1} has already been established (see Fischer et al., 2023, Theorem 23.11). An explicit formula is — to our knowledge — not yet known.
(Number of) trees with maximal value on \mathcal{BT}^\ast_n: Partially open problem. The number of binary trees T\in \mathcal{BT}^\ast_n with maximal maximum width for all n=2^h with h\in\mathbb{N}_{\geq 1} has already been established (see Fischer et al., 2023, Corollary 23.1). If n is not a power of two, the maximal tree(s) have — to our knowledge — not been analyzed yet.
Minimal value and (number of) trees with minimal value on \mathcal{T}^\ast_n and \mathcal{BT}^\ast_n: A lower bound for the maximum width for trees T\in \mathcal{T}^\ast_n which is tight for all n\in\mathbb{N}_{\geq 1} as well as the (number of) corresponding minimal trees have already been established (see Fischer et al., 2023, Theorem 23.12).
Expected value and variance under the Yule model: Open problem.
Expected value and variance under the uniform model: Open problem.