Let $G$ be a connected graph and $T$ a spanning tree of $G$. For $e \in E(G) - E(T)$, Let $C_{e}$ denote the unique cycle in $T + e$.

Let $H(T)$ be the the subgraph of $G$ induced by symmetric difference of of all the $C_{e}$ where $e \in E(G) - E(T)$. One could see that $E(G) - E(T) \subset E(H(T))$ and that $H(T)$ is an even subgraph.

My questions are:

1)Characterize the even subgraphs $K$ of $G$ such that there is a tree $T$ of $G$ and $H(T) = K$.

2) For which tree $T$ and $T'$ we have $H(T) = H(T')$?

3) If $G$ is even connected graph, is $G = H(T)$ for some tree $T$ of $G$? If not what other conditions needed for that to be true?