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We prove that there is a branch of the E‐tree which has an immediate successor to its minimum. This answers questions raised by D. Jensen and A. Ehrenfeucht in  and by H. Simmons in .
We review and generalize certain known facts about the intersections or projections of a given integral lattice with or into rational subspaces, and apply them to get a product formula for a numerical invariant associated to subspaces of Rn which are orthogonal to (1, 1,…, 1)ɛ Rn.
By number‐theoretical methods we give a result generalizing those of J. O. Shallit, one particular case of which reads as follows:
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