emat icians ▁from ▁New ▁York ▁( state ) ▁Category : S cient ists ▁from ▁New ▁York ▁City ▁Category : Vari ational ▁anal yst s ▁Category : M embers ▁of ▁the ▁United ▁States ▁National ▁Academy ▁of ▁Sciences <0x0A> </s> ▁In ▁type ▁theory , ▁containers ▁are ▁abstra ctions ▁which ▁permit ▁various ▁" collection ▁types ", ▁such ▁as ▁lists ▁and ▁trees , ▁to ▁be ▁represented ▁in ▁a ▁uniform ▁way . ▁A ▁( un ary ) ▁container ▁is ▁defined ▁by ▁a ▁type ▁of ▁shapes ▁S ▁and ▁a ▁type ▁family ▁of ▁positions ▁P , ▁indexed ▁by ▁S . ▁The ▁extension ▁of ▁a ▁container ▁is ▁a ▁family ▁of ▁dependent ▁pairs ▁consisting ▁of ▁a ▁shape ▁( of ▁type ▁S ) ▁and ▁a ▁function ▁from ▁positions ▁of ▁that ▁shape ▁to ▁the ▁element ▁type . ▁Cont ain ers ▁can ▁be ▁seen ▁as ▁canonical ▁forms ▁for ▁collection ▁types . ▁ ▁For ▁lists , ▁the ▁shape ▁type ▁is ▁the ▁natural ▁numbers ▁( including ▁zero ). ▁The ▁corresponding ▁position ▁types ▁are ▁the ▁types ▁of ▁natural ▁numbers ▁less ▁than ▁the ▁shape , ▁for ▁each ▁shape . ▁ ▁For ▁trees , ▁the ▁shape ▁type ▁is ▁the ▁type ▁of ▁trees ▁of ▁units ▁( that ▁is , ▁trees ▁with ▁no ▁information ▁in ▁them , ▁just ▁structure ). ▁The ▁corresponding ▁position ▁types ▁are ▁is omorphic ▁to ▁the ▁types ▁of ▁valid ▁paths ▁from ▁the ▁root ▁to ▁particular ▁nodes ▁on ▁the ▁shape , ▁for ▁each ▁shape . ▁ ▁Note ▁that ▁the ▁natural ▁numbers ▁are ▁is omorphic ▁to ▁lists ▁of ▁units . ▁In ▁general ▁the ▁shape ▁type ▁will ▁always ▁be ▁is omorphic ▁to