The 1-improper chromatic number of a graph. This is the least number of colours in a vertex-colouring, such that each vertex has at most one neighbour with the same colour.

The number of outer peaks of a permutation. An outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$ or $n$ if $w_{n} > w_{n-1}$. In other words, it is a peak in the word $[0,w_1,..., w_n,0]$.

The number of valleys of a permutation, including the boundary. The number of valleys excluding the boundary is [[St000353]].

The number of inner peaks of a permutation. The number of peaks including the boundary is [[St000092]].

The number of descents of a binary word.

The number of leaf nodes in a binary tree. Equivalently, the number of cherries [1] in the complete binary tree. The number of binary trees of size $n$, at least $1$, with exactly one leaf node for is $2^{n-1}$, see [2]. The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3].

The radius of a connected graph. This is the minimum eccentricity of any vertex.

Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra.

The number of occurrences of the contiguous pattern {{{[[.,.],[.,.]]}}} in a binary tree. Equivalently, this is the number of branches in the tree, i.e. the number of nodes with two children. Binary trees avoiding this pattern are counted by $2^{n-2}$.

The number of ascents of a binary word.