We present lower and upper bounds on adaptive heuristics for maintaining binary search trees using a constant number of link or pointer changes for each operation (constant linkage cost (CLC)). We show that no adaptive heuristic with an amortized linkage cost of o(log n) can be competitive. In particular, we show that any heuristic that performs f(n)=o(log n) promotions (rotations) amortized over each access has a competitive ratio of at least $\Omega(\log n/f(n))$ against an oblivious adversary, and any heuristic that performs f(n)=o(log n) pointer changes amortized over each access has a competitive ratio of at least $\Omega(\frac{\log n}{f(n)\log(\log n/f(n))})$ against an adaptive online adversary. In our investigation of upper bounds we present four adaptive heuristics: a randomized, worst-case-CLC heuristic randomized two-promotion (R2P) whose expected search time is within a constant factor of the search time using an optimal tree; that is, it is statically competitive against an oblivious adversary; a randomized, expected-CLC heuristic (locally optimized randomized partial splay (LORPS)) that has O(log n) expected-amortized update time and is statically competitive against an oblivious adversary; a deterministic, amortized-CLC heuristic (locally optimized partial splay (LOPS)) that has O(log n) amortized update time and is statically competitive against an adaptive adversary; a practical, randomized heuristic (randomized partial splay (RPS)) that is not CLC but has performance bounds comparable with those of the splay heuristic of Sleator and Tarjan; it is statically competitive against an adaptive adversary. The randomized heuristics use only constant extra space, whereas the deterministic heuristic uses O(n) extra space.