Spanning weakly even trees of graphs

Let $G$ be a graph (with multiple edges allowed) and let $T$ be a tree in $G$. We say that $T$ is $\mathit{\text{even}}$ if every leaf of $T$ belongs to the same part of the bipartition of $T$, and that $T$ is $\mathit{\text{weakly}}\mathit{\text{even}}$ if every leaf of $T$ that has maximum degree in $G$ belongs to the same part of the bipartition of $T$. We confirm two recent conjectures of Jackson and Yoshimoto by showing that every connected graph that is not a regular bipartite graph has a spanning weakly even tree.