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Source. Section 4.5, printed pp. 455–456 (published PDF).

Statement. An odd circuit has a unique maximum matching omitting any specified vertex. If an edge $e$ joins distinct outer vertices $v_1,v_2$ of a planted tree, the union of $e$ with the two root paths is a flower.

Proof. Deleting a specified vertex of an odd circuit leaves an even path. Its endpoint has only one possible matching edge, and successive deletion of these forced pairs gives its unique perfect matching. Adding back the omitted vertex gives a matching of half the remaining order, necessarily maximum.

Let $P_1,P_2$ be the tree paths from the exposed root $r$ to $v_1,v_2$. They are stems by Section 4.3. Their common part is a path from $r$ to a vertex $b$. The vertex $b$ is outer: if it were inner and the paths diverged there, it would have three incident tree edges, contrary to inner degree two. If one root path ends there, it is already one of the outer endpoints.

Each nonempty branch from $b$ to $v_i$ has even length, starts with a nonmatching edge and ends with a matching edge. The two branches are internally disjoint. Their union with $e$ is therefore an odd circuit whose only vertex unmatched by its internal $M$ edges is $b$. This remains valid when one branch has length zero. The common root path is a stem meeting that circuit only at $b$. $\square$