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We prove a functional central limit theorem stating that as $n\\to\\infty$ the process $$ D_n(u):= m^{\\frac 12 n} \\left(W_{\\infty}\\left(\\frac{u}{\\sqrt n}\\right) - W_{n}\\left(\\frac{u}{\\sqrt n}\\right) \\right) $$ converges weakly, on a suitable space of analytic functions, to a Gaussian random analytic function with random variance. Using this result we prove central limit theorems for the total path length of random trees. 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