Alternating Extremes in Graceful Labelings of Full Binary Trees and Spider Trees

We study a pinned form of graceful labeling. For full binary trees, we ask whether some deepest root-to-leaf path can carry the alternating extreme pattern $0,n-1,1,n-2,\dots$. Such a spine uses the extreme labels and largest differences, forcing all off-spine vertices and edges to use the middle labels and smaller differences, respectively. We prove this pinned-spine conjecture for comb full binary trees, verify it computationally for all rooted non-isomorphic full binary trees through order $23$, and give an example showing that a pinned-spine labeling cannot always be chosen as an $\alpha$-labeling. For spider trees, we prove a packing theorem for self-matched legs: pairwise disjoint legs based at hub label $1$, at least one of which contains label $0$, can be combined into a graceful spider, with unused labels attached as hub leaves. This yields graceful labelings for mixed-length spiders with sufficiently many leaves. We also report computations using a depth-first search ordered by largest unused differences and formulate the six-arm problem as an offset five-arm residual problem.