We study the fundamental problem of distributed network formation among mobile agents of limited computational power that aim to achieve energy balance by wirelessly transmi ing and receiving energy in a peer-to-peer manner. Speci cally, we design simple distributed protocols consisting of a small number of states and interaction rules for the formation of arbitrary and k-ary tree networks. Furthermore, we evaluate (theoretically and also using computer simulations) a plethora of energy redistribution protocols that exploit di erent levels of knowledge in order to achieve desired energy distributions among the agents which require that every agent has exactly or at least twice the energy of the agents of higher depth, according to the structure of the network. Our study shows that without using any knowledge about the network structure, such energy distributions cannot be achieved in a timely manner, meaning that there might be high energy loss during the redistribution process. On the other hand, only a few extra bits of information seem to be enough to guarantee quick convergence to energy distributions that satisfy particular properties, yielding low energy loss. * A preliminary version of this paper entitled "Peer-to-peer energy-aware tree network formation" appeared in Proceedings of the 16th ACM International Symposium on Mobility Management and Wireless Access (MOBIWAC), pages 1-8, 20181-8, [Madhja et al., 2018. is full version extends the conference one in multiple ways. It contains all missing proofs of the theoretical statements and additional details on the correctness and e ciency of our methods for the lossless case. Besides presenting protocols only for the formation of arbitrary and binary tree networks, we now further present protocols for the formation of k-ary tree networks, for any integer k ≥ 2, building upon ideas used for binary trees. For such networks, we also show that it is possible to achieve particular energy distributions when there is no energy loss, by designing new protocols for these cases as well. e evaluation of our methods in the lossy case has been extended to compare the e ciency of the protocols in terms of the energy that has been lost until a stable energy distribution has been reached. Our discussion on related work, generalizations of our model and open problems has also been extended signi cantly. Moreover, many examples have been included throughout the paper to illustrate how all proposed protocols work in particular cases. We would like to thank Dimitrios Tsolovos for fruitful discussions at early stages of this work that led to the publication of the preliminary conference version of this paper. is work has been partially supported by the Greek State Scholarships Foundation (IKY), and by the Alexander S. Onassis Public Bene t Foundation.