Crystallographic orientation was random in the range of initial solidification. Further, some grains are elongated along their direction. Columnar grains, whose growth directions are almost parallel to the heat flow direction.  Here, a dendrite whose growth direction is close to the heat flow direction overgrows the other dendrite whose growth direction is far from the heat flow direction. 

A grain is composed by columnar dendrite arms, which have the same crystallographic orientation and may be grown from one nucleus. After nucleation, each dendrite grows independently until it comes in contact with and impinges on adjacent growing dendrites. At this stage dendrite coherency has been reached, and the initial shape of the grains is formed.

See: Fundamentals of Solidification by Doru M. Stefanescu and Roxana Ruxanda, available on Researchgate.

Dear Nima Tabatabaei,

Dr. Florian Glodeanu has already answered your question.

Growth rate of dendrites will be the same as that of the grain because for this preferred orientation, growth rate of dendrites and solidification undercooling are minimized.

May be helpful: 

(1a) "Crystallographic investigation of grain selection during initial

solidification", H Esaka, Y Kataoka and K Shinozuk, 4th International Conference on Advances in Solidification Processes (ICASP-4), 2016. Please see the links if you have access:-

http://iopscience.iop.org/article/10.1088/1757-899X/117/1/012030/pdf

(1b) "COMPETITIVE GRAIN GROWTH IN DIRECTIONAL SOLIDIFICATION OF A NICKEL-BASE SUPERALLOY", Y.Z. Zhou, N.R. Green, TMS 2008: 

http://www.tms.org/Superalloys/10.7449/2008/Superalloys_2008_317_324.pdf

(2) Chapter on Aluminum Welding (page 487) in "Handbook of Aluminum: Vol. 1: Physical Metallurgy and Processes", edited by George E. Totten, D. Scott MacKenzie, 2003.  

(3)  Page 74:- 'on the formation of dendrites' in Book "PHYSICAL METALLURGY: PRINCIPLES AND PRACTICE", Third Edition,  by                     V. RAGHAVAN

Best wishes