can you rewrite my proof? **Proof:** 1. **Root the Tree**: Begin by rooting \(T\) at any vertex, structuring the tree from the root to leaves. This establishes a hierarchical framework for traversal. 2. **Edge Selection Process**: - **For Leaves**: Directly include the edge connecting any leaf labeled 1 to its parent in \(F\). This step ensures these leaves adhere to the odd adjacency condition. - **For Interior Vertices (Labeled 0)**: If connected to an odd number of edges in \(F\) via its children, skip adding the parent edge to keep the adjacency even. If connected to an even number, include the parent edge to maintain even adjacency. - **For Interior Vertices (Labeled 1)**: Include the parent edge if the current adjacency from children is even, ensuring odd adjacency. Skip if the adjacency is already odd. 3. **Ensuring Conditions at the Root**: The even total of labels across \(T\) ensures that the root's adjacency condition (even for 0, odd for 1) will naturally be satisfied through the selective process of adding edges to \(F\). 4. **Conclusion**: The edge selection strategy ensures a subset \(F\) of \(T\)'s edges meets the required conditions for all vertices, leveraging the tree's structure and the parity rule from the sum of labels. This approach guarantees compliance with the adjacency conditions for vertices labeled 0 and 1, rooted in the tree's acyclic nature and the initial even sum constraint