Distributing labels on infinite trees

Definition 4.2 (Mechanical tree). A tree is mechanical of density α ∈ [0; 1] if for all node v,there exists a phase φ v which satisfies one of the two following properties:⌋∀n : h v (n) =⌊S(n)α + φ v , (8)⌉or ∀n : h v (n) =⌈S(n)α − φ v . (9)In the first case, we say that φ v is an inferior phase of v. In the second case, we say that φ vis a superior phase of v.This definition suggests that the phases of all nodes could be arbitrary. In fact, we will seethat there exists a unique mechanical tree with a given phase at the root. The second questionraised by this definition is the existence and uniqueness of the phase: we call φ v “a” phase of anode φ v and not “the” phase of φ v since there may exist several phases leading to the same tree.We begin by a characterization of mechanical trees, given in the following formula:Proposition 4.2.1 (Characterization of mechanical trees). For each α ∈ [0; 1] and φ ∈ [0; 1),there exists a unique mechanical tree of density α such that φ is an inferior (resp. superior)phase of the root.Moreover, if φ is an inferior (resp. superior) phase of a node then φ 0 ≤ · · · ≤ φ d−1 areinferior (resp. superior) phases of its d children, withφ i =α + φ + i − ⌊α + φ⌋d(resp. φ v =)φ + ⌈α − φ⌉ − α + i. (10)dThe proof will be done in two steps. First we will see that if we define the phases as in (10)we have a mechanical tree, then we will see that this is the only way to do so.Proof. Existence. Let α ∈ [0; 1] and φ ∈ [0; 1). We want to build a mechanical tree whichroot has an inferior phase φ ( the case of a superior phase if similar and is not detailed here).Let A be an infinite tree. To each node v, we associate a number φ v defined by:• φ root = φ.• If the phase of a node v is φ v , its d children satisfy Equation (10).Then we build a labeled tree by associating to each node v the label ⌊α + φ v ⌋. Let us prove byinduction on n that the following relation holds.⌋For all v : h v (n) =⌊S(n)α + φ v . (11)By definition of the <strong>labels</strong>, (11) holds when n = 1. Let n ≥ 0 and let us assume that (11) holdsfor n. Let v be a node with phase φ v and let φ 0 . . . φ d−1 be the phases of its children. We assumethat α + φ v < 1 (which means that the label of the node is 0) a similar calculation can be donein the other case, α + φ v > 1.13