We generalize the Hermite-Korkin-Zolotarev (HKZ) reduction theory of positive definite quadratic forms over Q and its balanced version introduced recently by Beli-Chan-Icaza-Liu to positive definite quadratic forms over a totally real number field K. We apply the balanced HKZ-reduction theory to study the growth of the g-invariants of the ring of integers of K. More precisely, for each positive integer n, let O be the ring of integers of K and g(O)(n) be the smallest integer such that every sum of squares of n-ary O-linear forms must be a sum of g(O)(n) squares of n-ary O-linear forms. We show that when K has class number 1, the growth of go (n) is at most an exponential of root n. This extends the recent result obtained by Beli-Chan-Icaza-Liu on the growth of g(Z)(n) and gives the first sub-exponential upper bound for g(O)(n) for rings of integers O other than Z.