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The longstanding graceful tree conjecture, posed by R\\'{o}sa in the 1960s, asserts that every tree is graceful. The $\\textit{graceful}$ of an $n$-vertex tree $T$, denoted $\\operatorname{gs}(T)$, is the maximum possible number of distinct edge-differences over all bijective labellings $\\phi:V(T)\\to \\{1,\\ldots,n\\}$. 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The longstanding graceful tree conjecture, posed by R\\'{o}sa in the 1960s, asserts that every tree is graceful. The $\\textit{graceful}$ of an $n$-vertex tree $T$, denoted $\\operatorname{gs}(T)$, is the maximum possible number of distinct edge-differences over all bijective labellings $\\phi:V(T)\\to \\{1,\\ldots,n\\}$. 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