which are conditional on the Galton-Watson tree. Section \[sec:prelim\] culminates in Lemma \[lem:X\] and Corollary \[cor:X\]. Section \[sec:g\] gives an upper bound on survival probabilities and squeezes survival probabilities close to an easier-to-analyze averaged version with high probability. Section \[sec:backbone\] proves two Markov properties for the subtree from $\gamma_n$ together with $\pivot (\gamma_n)$. The remainder of the section extends the work of [@angel-invasion2008] by proving a limit law for $\pivot (\gamma_n)$ which then implies an upper bound on the rate at which $\pivot (\gamma_n) \downarrow p_c$. In particular, Corollary \[cor:LPE\] shows convergence to the Poisson lower envelope process, as in [@angel-invasion2008]. Section \[sec:rest\] proves Theorem \[th:main\] by comparing the conditional invasion measure to the ratio of survival probabilities and by carefully estimating survival probabilities near criticality.

A glossary of notation by page of reference is included after the references.

Construction and preliminary results {#sec:prelim}
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Galton-Watson trees
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We begin with some notation we use for all trees, random or not. Let $\ulam$ \[ulam-def\] be the canonical [*Ulam-Harris*]{} tree [@louigi-ford]. The vertex set of $\ulam$ is the set $\VU : = \bigcup_{n=1}^\infty \N^n$, \[VU-def\] with the empty sequence $\rtt := \emptyset$ \[rtt-def\] as the root. There is an edge from 