Bijection: tree-like lists - natural numbers
Python 2, 118 114 109 105 103 97 94 bytes



f=lambda x:x&gt;[]and f(x[:-1])*2+1&lt;&lt;f(x[-1])
g=lambda n:[g(len(w))for w in bin(n).split('1')[1:]]


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Same encoding as the example...