( are true to justify these claims note first thatby the size propertywe can have at most nodes at depth at most nodes at depth and so on thusthe number of external nodes in is at most likewiseby the depth property and the definition of ( , treewe must have at least nodes at depth at least nodes at depth and so on thusthe number of external nodes in is at least in additionby proposition the number of external nodes in is thereforewe obtain <= + and + <= taking the logarithm in base of each of the above termswe get that <=log( + and log( + <= hwhich justifies our claims ( and proposition states that the size and depth properties are sufficient for keeping multi-way tree balanced (section moreoverthis proposition implies that performing search in ( , tree takes (logntime and that the specific realization of the secondary structures at the nodes is not crucial design choicesince the maximum number of children max is constant ( we canfor exampleuse simple ordered dictionary implementationsuch as an array-list search tablefor each secondary structure update operations for ( , trees maintaining the size and depth properties requires some effort after performing insertions and removals in ( , treehowever we discuss these operations next insertion to insert new entry ( , )with key kinto ( , tree twe first perform search for assuming that has no entry with key kthis search terminates