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The standard relay-contact-tree network has been used extensively for many years. If n is the number of relays involved, it has always been assumed that the 2(2n−1) contacts used in the standard tree network is the smallest possible number of contacts with which such a network could be made. This paper proves that this is true, provided no sneak paths are allowed. This is in contrast to the result obtained by Lupanov, who showed that when n is five or more it is possible to save contacts below the usual number by permitting sneak paths. This paper proves further theorems about any network which satisfies the same specifications as an n-relay tree without sneak circuits, and which is built with the minimal number of contacts. In particular, these theorems characterize such a network well enough that it can be shown to be one of the standard forms of relay tree network.
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