Distributing labels on infinite trees

Balanced TreesRational Trees8IrreducibleSturmian Trees435Mechanical TreesStrongly balancedUltimatelyMecha7621Reducible9Figure 14: Relations of inclusion linking the different definitions that we presented. Each numberrefers to an example detailed in section 7. For example 5 is the set of trees that are rational,reducible, ultimately mechanical, strongly balanced, balanced and neither mechanical nor Sturmian.0 1 2 3 4 5 ...Figure 15: The reflected random walk tree: each node of type n is followed by one of type n − 1and one of type n + 1 (except for 0 that is followed by 0 and 1).3. Irreducible rational trees – see Figure 6.4. Reducible rational trees – see Figure 5.5. Rational reducible strongly balanced tree that is not mechanical – strongly balanced treeare not necessarily mechanical in the case of reducible rational trees but only ultimatelymechanical, see Figure 8 for an example.6. Reducible mechanical trees – let α be a normal number and consider the mechanical tree ofdensity α and phase 0 at the root. As α is normal, there is a unique phase correspondingto each node of order k which is the fractional part of:α( 1 d k + · · · + 1 d ) + i kd k + · · · + i 1dfor a unique sequence i 1 , . . . , i k . One can show that if two sequences of i 1 , . . . , i k aredifferent, then these phases are different, which shows that the minimal graph of the treeis exactly the tree itself.(26)26