The depth or height of a binary tree. The depth (or height) of a binary tree is the maximal depth (or height) of one of its vertices. The '''height''' of a vertex is the number of edges on the longest path between that node and a leaf. The '''depth''' of a vertex is the number of edges from the vertex to the root. See [1] and [2] for this terminology. The depth (or height) of a tree $T$ can be recursively defined: $\operatorname{depth}(T) = 0$ if $T$ is empty and $$\operatorname{depth}(T) = 1 + max(\operatorname{depth}(L),\operatorname{depth}(R))$$ if $T$ is nonempty with left and right subtrees $L$ and $R$, respectively. The upper and lower bounds on the depth of a binary tree $T$ of size $n$ are $log_2(n) \leq \operatorname{depth}(T) \leq n$.

The minimal length of a chain of small intervals in a lattice. An interval $[a, b]$ is small if $b$ is a join of elements covering $a$.

The largest part of an integer partition.

The maximal part of the shifted composition of an integer partition. A partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ is shifted into a composition by adding $i-1$ to the $i$-th part. The statistic is then $\operatorname{max}_i\{ \lambda_i + i - 1 \}$. See also [[St000380]].

The maximum of the length and the largest part of the integer partition. This is the side length of the smallest square the Ferrers diagram of the partition fits into. It is also the minimal number of colours required to colour the cells of the Ferrers diagram such that no two cells in a column or in a row have the same colour, see [1]. See also [[St001214]].