The rank of the endomorphism monoid of a levelwise regular tree

The rank of a monoid $M$ is the minimal number of generators of $M$. We say that a tree of height $k$ is levelwise regular of type $(m_1,\dots,m_k)$ if the number of children of every vertex on level $i$ is $m_i$. Under the assumption that all $m_i\ge 2$ we show that the rank of the endomorphism monoid of a levelwise regular tree of type $(m_1,\dots,m_k)$ is $2k$.