We improve the analyses for decision tree boosting algorithms.
Our result consists of two parts.
In the first part, we analyze a decision tree learning algorithm
proposed by Takimoto and Maruoka that produces a $K$-way branching decision tree
(where $K \geq 2$ is a given constant) for multiclass classification.
Under the assumption that 
the algorithm can always obtain a weak hypothesis with advantage $\geq \gamma$,
Takimoto and Maruoka showed that
$(1/\varepsilon)^{(K/\gamma)}$ boosting steps are sufficient
for obtaining a tree with the desired training error $\varepsilon$.
We improve this bound
by proving that $(1/\log K)(1/\varepsilon)^{(\log K/\gamma)}$ steps are indeed sufficient.
In the second part,
we study a multi-way decision tree learning algorithm
proposed by Mansour and McAllester
that produces a tree with nodes having different number of branches.
By using a simpler analysis than the one given in the original result,
we simplify and improve their algorithm.