and bloga+logb < logc- ( let us consider the change in (tcaused by each type of splaying substep zig-zig(recall figure since the size of each node is one more than the size of its two childrennote that only the ranks of xyand change in zig-zig operationwhere is the parent of and is the parent of alsonote that ( )+ '( < '(xthusby ( )+ '( < '( )- that isr '( < '( )- ( )- this inequality and imply < '( )+( '( )- ( )- )- ( < ( '( )- ( ))- zig-zag(recall figure againby the definition of size and rankonly the ranks of xyand changewhere denotes the parent of and denotes the parent of alsor '(xr(zand ( < (ythus '{xr '{yr '{ )- { )- { )- { < '(yr '( )- ( )- ( < '(yr '( )- ( ( note that '(yn '( < '( )henceby '( )+ '( < '( - thusd < '( )- - ( < ( '( )- ( ))- zig(recall figure in this caseonly the ranks of and changewhere denotes the parent of alsor '(yr(xthus '( )+ '( )- ( )- (