$($Keep four digits after the decimal point in calculation and round final answer to two places$)$

Future stock prices are modeled with a $1-$period binomial tree, the period being $6$ months.

$($i$)$ The stock's current price is $20.$

$($ii$)$ The continuously compounded risk $-$free interest rate is $3\%.$

$($iii$)$ The stock pays continuous dividends proportional to its price at a rate of $1\%.$

$($iiii$)u=1\mathrm{.}1,d=0\mathrm{.}9$

$(1)$ Construct the binomial tree for a European call option on the stock expiring in $6$ months with a strike price of $20.$

$(2)$ What is the number of shares $()$ and B in the replicating portfolio for a European call option on the stock expiring in $6$ months with a strike price of $20?$

$(3)$ What is the risk$-$neutral probability $P^{**}?$ Use $P^{**}$ to find the no$-$arbitrage premium for the European call option on the stock expiring in $6$ months with a strike price of $20.$

$(4)$ The market price of the European call option on the stock expiring in $6$ months with a strike price of $20$ is $$1\mathrm{.}20.$ John created a portfolio including buy one share of the call option sell $()$ shares of underlying and sell B bonds. Find John's profit for node $u$ and node $d$ separately.