The minimum size of a contiguous k-universal d-ary plane tree is exactly d^{k-1}+k-1; the noncontiguous variants have minimum sizes between roughly k log_2 k and k^{(1/2) log_2 k}.
Butler, Induced-universal graphs for graphs with bounded maximum degree
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The minimum size of a contiguous k-universal d-ary plane tree is exactly d^{k-1}+k-1; the noncontiguous variants have minimum sizes between roughly k log_2 k and k^{(1/2) log_2 k}.