  The tangent number $T_{2n+1}$ is equal to the number of increasing labelled
complete binary trees with $2n+1$ vertices. This combinatorial interpretation
immediately proves that $T_{2n+1}$ is divisible by $2^n$. However, a stronger
divisibility property is known in the studies of Bernoulli and Genocchi
numbers, namely, the divisibility of $(n+1)T_{2n+1}$ by $2^{2n}$. The
traditional proofs of this fact need significant calculations. In the present
paper, we provide a combinatorial proof of the latter divisibility by using the
hook length formula for trees. Furthermore, our method is extended to $k$-ary
trees, leading to a new generalization of the Genocchi numbers.
