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Source. Section 7.2, printed pp. 465–466 (published PDF).

Statement. Suppose a retained pseudovertex $b'$ is inner in a planted tree $T'$ for a quotient matching $M'$. Expand its remembered odd circuit $B$. Let $b_1$ be the attachment in $B$ of its matching tree edge, and $b_2$ the attachment of its nonmatching tree edge. Replace $b'$ in the tree by the even arc of $B$ between these vertices, using the zero-edge arc if $b_1=b_2$. The resulting graph is a planted tree for the lifted matching.

Proof. The inner vertex has exactly two tree edges, one matching and one nonmatching. Lift $M'$ by the unique circuit matching $M_B$ omitting $b_1$, as in Section 4.14. If $b_1\ne b_2$, the two arcs between them have opposite parity, since $B$ is odd. Along either arc from $b_1$, the edges of $M_B$ alternate, beginning with a nonmatching edge. Thus the even arc ends with a matching edge at $b_2$.

Replacing the original degree-two vertex by this path leaves a tree. Label its endpoints $b_1,b_2$ inner, and alternate the labels along the even arc. All new inner vertices have degree two, counting the two external tree attachments at its ends. The matching and nonmatching edges alternate through the replacement: the external matching edge covers $b_1$, and the internal final matching edge covers $b_2$.

The unused arc's interior is even in size and is matched internally by $M_B$. No edge of the lifted matching crosses from that interior into the replacement path. Thus the enlarged tree retains its original exposed root and is planted in the expanded graph.

If $b_1=b_2$, use just this one vertex with its two external tree edges. All other circuit vertices are matched internally, and the same conclusions hold. Any child pseudovertices still denote their unchanged blocks and edge attachments; they may be expanded later in the same way. $\square$