DENSITY BOUNDS FOR THE 3X+1 PROBLEM .1. TREE-SEARCH METHOD

The 3x+1 function T(x) takes the values (3x+1)/2 if x is odd and x/2 if x is even. Let a be any integer with a not equal 0 (mod 3). If n(k)(a) counts the number of n with T-(k)(n) = a, then for all sufficiently large k, (1.302)(k) less than or equal to n(k)(a) less than or equal to (1.359)(k). If pi(a)(x) counts the number of n with \n\ less than or equal to x which eventually reach a under iteration by T, then for sufficiently large x, pi(a)(x) greater than or equal to x(.65). The proofs are computer-intensive.