Asymptotic costs confusion -

so i'm trying head around different asymptotic costs, can tell faster can't grips when occur /why, can explain me when of examples below occur? or other complexities

knowing how complexity occurs largely matter of understanding factors arise , putting them in way. if operations being performed done in correlated manner other operations, complexities multiplied; if performed in series complexities added.

list traversals, example, each order n; might know right off bat o(n²) complexities arise doubly nested loop algorithm, o(n³) triply nested one.

now writing complexities multiple terms not, in experience, common practice - o(n² + 2n + 1) example. because typically care complexities large n , in such case, lower order terms become negligible. however, can visualize such thing nested loop algorithm in 2 separate steps performed in outer loop that's o(1) (maybe adding collection), plus single o(1) operation outside of loops.

similarly, tree structures yield logarithmic complexities. height of complete , balanced binary tree proportional log₂(n), n number of nodes in tree. thus, finding in tree using recursive algorithm - can thought of stepping through tree's levels - log₂(n).

note in worst case binary tree isn't complete , balanced (ie. each node has 1 child filled in), tree topologically identical list, , searching through o(n). that's why many algorithms list range of complexities.