What determines how fast a child grows in height in one year?

Growth rate. Height velocity varies with age, caloric intake, hormone levels, and absence or presence of disease. Your pediatrician can plot a growth velocity for your child and show you how normal she is.
Diet and hormones. Growth is modulated by growth hormone primarily. Good nutrition and growth hormone are the big determinants for growth. The first year of life and puberty are the two times that growth is the fastest. Most children grow 2-4 inches a year.