We consider random finite $k$-regular trees for any fixed integer $k\ge2$.
We consider a branching process such that a node has $k$ children with probability $p_k$, and it has no child with probability $1-p_k$.
Let $T$ denote a tree generated by this process.
Let the height of a node in $T$ be the distance from the node to the root of $T$ and let the level of a node in $T$ be the distance form the node to the farthest descendant in $T$.

The height of $T$ as well as the number of nodes with a given height $h$ have been studied in depth in the literature.
On the other hand,
less is known
about the number of nodes of given level $l$.
The purpose of this note is to show
reasonable upper and lower bounds for
the expected number of nodes of given level $l$.